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0, &k < 00, and nk are to be chosen in such a way that there
III. Norm Convergence of Fourier Series
64
is an appropriate interplay between the (5.16) holds for J; note that
&k'S
and the Ink's. To show that
It only remains now to choose the &k'S and nk's so that the right-hand side of (5.22) goes to infinity with k; the reader can verify that nk = 2(k!)2 &k = 1/k!, will do. Therefore J k + CO as k + 00 and (5.16) holds for a function f of period 1. The desired result follows without difficulty from this. We need one more notation before we go on. By the convolution formula (3.3) of Chapter I we readily see that if
(5.23) then also
1I,
G,,(f; X ) = 21Zn * f ( x ) = Moreover, since
f(X
- t)k,,(t) dt.
(5.24)
k,,(t)is odd (5.24) can be rewritten as either (5.25)
Furthermore, since the integrand in the last integral of (5.25) is even, we also have
(5.26)
65
6. The Truncated Hilbert Transform 6. THE TRUNCATED HILBERT TRANSFORM
Proposition 5.1 motivates the consideration of a new integral operator, namely, the truncated Hilbert transform H E f which ; is given by an absolutely convergent integral and which we expect will converge, in some sense, to f: Definition 6.1. For f E L( T) and 0 < E < n-, let
As usual, we do the L2 case first. Proposition 6.2. Suppose f
E
-fll~
LZ(T).Then lirn,,,+IlH,f
= 0.
Proof. We begin by showing that
IIKf 112 6 cllf 112 with c independent of E and f: In first place observe that 1 2tan(t/2)
HEf(X) = -
+'I n-
E 0, say, then by 7.19 ;,,(A x) + -00, and again by FejCr's theorem a,(X x) = &,,(A x) + -00.) 7.21 (Localization) Iff vanishes in an interval Z c T, then ;,,(A x) converges uniformly in any subinterval of the interior of I. (Hint: It is a straightforward variant of 5.8 and 5.9 of Chapter 11.) 7.22 The distribution p.v. (;tan(t/2)) has Fourier coefficients c k = ( - i ) sgn k (Hint: Apply (5.14) to f ( x ) = e 7.23 If a real sequence bk decreases to 0, then the trigonometric series bk sin kt converges everywhere. Moreover, for any 6 > 0 the series converges uniformly in 6 < 1x1 < T. (Hint: As in 5.8 of Chapter I use Abel's summation formula and estimate (5.5) this time. Since all sine terms vanish when t = 0 the series converges now everywhere.) The question of the uniform convergence of the above series must be decided in a different manner. 7.24 If a real sequence bk decreases to 0,then for the uniform convergence of 1bk sin kt in T it is necessary and sufficient that kbk + 0 as k +. 0O. (Hint: The necessity follows at once upon observing that l ~ ~ ~ mbk+ sin l kxl can be made uniformly small and letting x = rr/4m there. As for the sufficiency, on account of 7.23 it is enough to show that, if r , ( t ) = bk sin kt and E , = maxkankbk, then Ir,( t)l S CE, for tin [ O , ~ / 4 ] Since . it is possible to find an integer N such that 1/ N < t 1 / ( N - l), then write I".)
c';p=,
N-1
r n ( t )=
C+C
(k=n
bk sin(kt) = rn,l(t)+ rn,z(t)>
k:N)
say, where r,.l(t) = 0 if N s n. To estimate r,.l(t), use that [sin ktl to estimate r,J t ) , distinguish two cases.)
S
kltl;
7. Notes; Further Results and Problems
73
There exists a trigonometric series that converges uniformly but not absolutely in T. (Hint: Cz=’=,(sin(kt))/(kIn k).) 7.26 If bk is a real sequence decreasing to 0, and kbk is bounded, then the partial sums of the trigonometric series ITzl bk sin(kt) are bounded. In particular IC;=,(sin(kt)/k)l 6 C, all t. 7.27 If the real sequence bk decreases to 0, the condition kbk + 0 is necessary and sufficient for C bk sin( kt) to be the Fourier series of a continuous function. (Hint: It is enough to prove the necessity of the condit) of the above series converge uniformly, then tion; if the (C, 1) means a,,( un(7r/2n) + 0 and 7.25
The result follows without much difficulty from this.)
CHAPTER
IV The Basic Principles
1. THE CALDERON-ZYGMUND INTERVAL DECOMPOSITION
In this section we discuss one of the most important topics of harmonic analysis, namely, the Calderbn-Zygmund decomposition of an integrable function. This is a principle which roughly states that an integrable function f can be written as a sum of two functions, f = g b say, where the “good part” g is essentially bounded and the “bad part” b has a cancellation property and bounded averages over a particular collection of open, disjoint subintervals of T. Let us be more precise. In first place we assume that f is a nonnegative function, for otherwise we can apply the result to to include arbitrary real- or complex-valued functions as well. Let then A > 0 be a number exceeding f T , the average off over T, i.e.,
+
In
We will work with the family of open, dyadic subintervals of T, i.e., those intervals obtained by successively subdividing T into equal, open subintervals. The first subdivision of T thus consists of TI = (-‘IT, 0) and T2 = (0, ‘IT). Observe that since
then at least one of the averages f T , or f T ; does not exceed A. Also,
74
1. The Caldero'n-Zygmund Interval Decomposition
75
so that even if an average exceeds A, it remains bounded by 2A. We can now get the selection process for intervals going. In case the average off over a subinterval does not exceed A, then we continue subdividing that interval. If, on the other hand, the average off over an interval does exceed A, then we separate that interval and rename it I , . Clearly,
Suppose, then, that after k steps Z,,12,.. . ,Z, have been separated. The intervals {I}which have not been renamed and separated at that step have the property that h < A. We therefore let each Z in this collection play the role of T above and subdivide it into two equal, open subintervals. The average off over either of these intervals does not exceed 2A, and over at least one of them it does not exceed A. Those subintervals of {I}for which . . ,I,,,, the average off exceeds A and is bounded by 2A are renamed Z,,+,,. say, and separated. These are the intervals obtained in the (k + 1)-step. This selection procedure thus produces a collection of open, disjoint, dyadic subintervals (4) of T with the following properties: 1
f ( y ) d y A } = {Mf>A}, is open for every A > 0. Indeed, let { x , } be a sequence of points in T\OA which converges to x ; we show that x E T\O, as well. It clearly suffices to show that for every interval I = (x - 7, x + T ) , 0 < T S T,we have
Let In = ( x , - T , x n + T ) and put fn(Y) = ~ ( Y ) X ~ , ~ I ( where Y), I,, AZ = ( Z n \ I ) u (Z\I,,)is the symmetric difference of I,,and I. Since Ifn(y)l s If(y)l and f,,(y) + 0 as n + m, by the Lebesgue dominated convergence theorem it follows that
Moreover, since x , E T\OA
77
2. The Hardy-Littlewood Maximal Function Now, by (2.2),
Thus letting n + 00 above, and on account of (2.1), we obtain that (1/(11) j I ( f ( y ) ldy s A, which is precisely what we wanted to show. So Mf(x) is a positive, measurable function, but is it integrable? A simple example shows that this is not necessarily the case. Indeed, let f(y) = x(o,,,z)(y)(d/dy)(l/ln(l/y)). Then f(y) 3 0 and f E U T ) . Now for x in (-4,O) we have Mf(x) 3
2.11
I,
f(v) dy,
all
77
> 0-
x- rl,x+rl)
In particular setting 77
= 21x1,
we get
and this function is not integrable in a neighborhood of 0. To deal with this inconvenience we introduce the weak-L class of Marcinkiewicz. We say that a measurable function f ( x ) is in wk-L( T) provided there is a constant c such that AIb
E
T : If(x)l> A l l = Al{lfl>
A l l S c,
all A > 0.
(2.3)
The infimum of the constants c appearing in (2.3) is called the "wk-L norm" off; although this quantity does not satisfy the usual requirement of a norm. By Chebychev's inequality we see that integrable functionsf are in wk- L( T), with norm not exceeding llflll. On the other hand the function f ( x ) = (l/x)(,q,,l)(x)) is in wk-L( T), although it is not even locally integrable in a neighborhood of 0. We can now prove the Hardy-Littlewood maximal theorem, namely, Theorem 2.1. Suppose f E L( T). Then Mf E wk-L( T) and the wk-L norm of Mf does not exceed 72rrllfll,. More explicitly A ({Mf > A}[
G
36
[
If(y)I dy,
all
A > 0.
(2.4)
T
Proof. Since Mf(x) = M(lfl)(x), we may assume that f is a real-valued, nonnegative function. Moreover, since hI{Mf> A}l 6 A ~ T all , A, we may also assume that ( 1 / 2 ~ j)T f ( y )dy < h/18, for otherwise there is nothing to prove.
ZV. The Basic Principles
78
Let then fT < A/18, and invoke the Calder6n-Zygmund interval decomposition at level A/18. We then have the collection (1,) of open, disjoint, dyadic subintervals of T as well as the functions g and b,f = g + b, verifying properties (1.1)-(1.7). Now, it is clear that M is a sublinear mapping, i.e., M ( f , + f z ) ( x ) s M f , ( x )+ Mf2(x),and, consequently,
M f ( x ) s M g ( x )+ M b ( x ) ,
all x E T.
(2.5)
For each of the intervais 4, let 21, denote the concentric interval with 4 with measure (or length) twice that of 1,. Finally, put = 21,. With the aid of (2.5) we will be able to show that
a* uj
Mf(x)s A,
XE
T\a*.
(2.6)
If this is the case, then {Mf> A} c a* and by (1.2), Al(Mf> A } \ S h C 1 2 1 , 1 = 2 A C l l i l i
i
and we are done. It thus remains to prove (2.6). Let Z be an interval centered at x E T\a*; we begin by showing that
This is not hard. Since b = 0 off a, we have that
where the sum extends only over those j’s with Z n 1, # 0.We divide these 4 ’ s into two mutually exclusive families, namely, (i) IZ n 1,l 3 11,1/2, we call these intervals Z(’)’s, and (ii) IZ n 41 < 11,\/2, we call these intervals Z(’)’s. For the Z‘”’s we immediately see that
2. The Hardy-Littlewood Maximal Function
79
For the Z(2)'swe invoke the fact that x E T\n*. Since in particular x iZ 24, allj, we have [ I n 41 2 141/2. Thus we see that
=s 16(h/18)1I n $1.
Whence collecting estimates (2.8) and (2.9) we obtain
which is precisely (2.7). Since I is arbitrary in (2.7) this means that M b ( x ) s 16h/18,
x
E
T\n*.
(2.10)
I, g ( y ) dy.
Again for an interval I centered at x, we estimate now (1/111) By the definition (1.4) of g we readily see that
I,
g ( Y ) dv
=
c
I I n 41
jr,m + j p dY
dY = A1 + A2,
j.1 n 1,# 0
say. Also from (1.1) it readily follows that (2.11)
To estimate A2 we again resort to the geometry of the situation. In first place it will suffice to consider the integral extended to the open set I\uj 4 c I. This set can be written as a countable, disjoint union of open intervals J k , say. Each of the Jk's such that Jk does not contain an endpoint of I (and there may be only two of these at that) is, in turn an a.e. union of disjoint, dyadic subintervals Jkn,say, of T which were subdivided in the Calder6n-Zygmund decomposition process. Thus by (1.l), (2.12) On the other hand, if .&,, contains an endpoint of I, let n be the integer ~ , the such that (27~/2"+')=s lJkl< (27~/2").Then J k c J ~ u, J ~where are disjoint, dyadic subintervals of T, each of measure s27r/(2n + l), which
ZV. The Basic Principles
80
were not separated in the Calderbn-Zygmund decomposition process. Thus we have
=z
WWIJk,l+
IJk21)
< (A/18)21JkI.
(2.13)
Combining the estimates (2.12) and (2.13), we readily see that (2.14)
Thus adding (2.11) and (2.14) we get
s 2(A/l8)lZl.
Since Z is arbitrary from this estimate it follows that M g ( x ) s 2(A/18),
all x
E
T.
(2.15)
Finally, by estimating the right-hand side of (2.5) by (2.10) and (2.15), we obtain that
M f ( x ) s 16A/18
+ 2A/18 = A,
x E T\R*,
which is precisely (2.6). An interesting corollary to the maximal theorem is a version of the Lebesgue differentiation theorem, which we hope will be useful in estimating the right-hand side of (1.3). Observe, however, that what we really n5ed is a “noncentered” version of our results. Let us do this then. Let M f ( x ) denote the noncentered maximal function defined by
r x. It is trivial to where the Z’s are open intervals of length ~ 2 7 containing see that l@f is lower-semicontinuous and therefore meas_urable. Moreover, the Hardy-Littlewood maximal theorem also gives that Mf E wk-L( T) with norm s2167r11fl11. This is easy to see since for each open interval Z containing x the interval I, = (x - IlZl, x $111)contains Z. Therefore
+
2. The Hardy-Littlewood Maximal Function
81
and, consequently, A&x) s 3 M f ( x ) . Thus {A@> A} = { M f > A / 3 } and the desired estimate follows at once from (2.4). We can now state, and prove
Theorem 2.2 (Lebesgue Differentiation Theorem). Suppose that f E L( T ) . For each x E T let $x = {I,}be a family of open intervals containing x, converging to x as (Z,I + 0. Then
'I
lim I L l + O I,[
I
f(y) dy = f ( x )
a.e. in
T.
I,
Proof. With no loss of generality we may assume that f is real valued. It is also obvious that the desired conclusion holds true for a trigonometric polynomial, or even a continuous function J Such functions are dense in L ( T ) , and we may assume that they are also real-valued when f is (cf. Remark 2.5 of Chapter 11). Let
Clearly, + ( A x ) = +(f - p , x ) for every real-valued trigonometric polynomial p. Moreover, since +(g, x ) s 2&(x), we also have that 4(J;x) s 2 @ ( f - p ) ( x ) , all x in T. Thus for each A > 0, {+(f)> A } = { M ( f - p ) > A/2}, and by Theorem 2.1
I{d(f) > A } I
(4A)ll.f-Plll.
(2.16)
But since the right-hand side of (2.16) can be made arbitrarily small it follows that > A}[ = 0, each A > 0. This in turn implies that for a.e. x in T
I{+(f)
1 f(y) dy = lim inf and, consequently, lim~Ix~+o(l/~Ix~) j I x f ( y )dy exists for almost all x E T. It is now immediate to see that this limit is actually f ( x ) a.e. Indeed, for A > 0 Put
Now, again for an arbitrary trigomometric polynomial p, we have
IV. The Basic Principles
82 Therefore
QfO)= { G ( f - P )> = %,-I -k
u {IP
-A
>
%A27
say. By choosing p appropriately, as was done above, we see at once that 1%,-1 can be made arbitrarily small. But a similar conclusion holds for %x2 since by Chebychev’s inequality
1421; Whence I%,(A)l
j T
I f ( x ) - P ( X ) l dx = 2 4 f - P l l l .
= 0 for each A
> 0 and
‘I
lim - f ( y ) dy = f ( x ) , a.e. in T. W I I,] 1, Theorem 2.2, in one of its simplest formulations can be restated as folows. Let X ( t ) denote the characteristic function of the interval (-4,f) and let X E ( f ) = (2T/E)X(t/E). Then IL-0
In particular, iff is nonnegative (and in the general case also by replacing s u p o < E < 2 *~xE)I ~ f S 6 f f ( x ) .Thus the convolution with the kernal ~ ~ ( t )which , roughly represents a “bump” at the origin of width E, height ~ T / Eand , total mass 1, is controlled by the Hardy-Littlewood maximal operator, and the differentiation of the integral, namely, statement (2.17) holds a.e. in T. The natural questions then is, under what general conditions on an arbitrary kernel will (2.17) remain valid. The particular examples we have in mind are those kernels which look like the Fejer K, kernels; also note that the story is totally different for the Dirichlet kernels 0,since they have variable sign and more importantly IID,II, + 00 as n + 00. More precisely, then, for an integrable function on the line R, under that conditions does
f by If\),
+
(2.18) for x a.e. in T? A closer look at the proof of Theorem 2.2 indicates that a positive answer to (2.18) depends upon the following two properties of (i) sup,lj,f(x - t ) n + ( n t ) dtl E wk-L( T) whenever f~ L( T); $ ( t ) dt)f(x), for a dense class of (ii) limn+mJ , f ( x - t ) n + ( n t ) dt = smooth functions f E L( T ) .
+:
(I,
2. The Hardy-Littlewood Maximal Function
83
+
if we restrict ourselves to (ii) requires no further assumption on trigonometric polynomials. In fact, by linearity, we may assume that f ( t ) = e"', and in this case
I
e"x-')n+(nt) dt = e"
(-m,TrI
1.
X ~ - n + r , n ~ , ( t ) e - i i f / dt. n+(t)
If we denote by +,( t ) the integrand of the above integral it readily follows that I+,(t)l s 1+(t)l and limn+m& ( t ) = + ( t ) a.e.; consequently, by the Lebesgue dominated convergence theorem (ii) holds. To prove (i) it actually suffices to show that the sup in question is majorized by a wk-L function; a constant multiple of M f ( x ) will do. This requires the following argument.
+
Proposition 2.3. Let be an integrable function in R which admits a nonincreasing, even, integrable majorant T ; more precisely, I+(t)l =sd t ) ,
where 7 is even,
I,
t E R,
7 ( t )dt < 00 and ~ ( s z) v ( t ) , 0 < s < t. Then
Proof. Clearly,
m
r
1n 7 ( n 7 ~ / 2 ~ + ' ) k=O
m
I f ( x - t)1 dt
lrl=?r/2*
=
Ak,
(2.19)
k=O
say. It is also readily seen that the integral in Ak is bounded by
and, consequently, Ak s ( 2 7 ~ / 2 ~ ) n ~ ( n r r / 2 " " ) M f ( x ) all ,
k, n, x.
Thus summing over k we get (2.20)
ZV. The Basic Principles
84
Moreover, since
I
tq(t)s 2
q ( s ) ds,
t 20,
(t/2*11
the sum on the right-hand side of (2.20) is bounded by
8 f I &=O
q ( s ) ds s 4
(fl?r/zk+',fl?r/
I,
q ( s ) ds,
(2.21)
2'1
independently of n. Our conclusion follows combining (2.19)-(2.21). Corollary 2.4. Under the assumptions of Proposition 2.3, for f
lim
n-tm
I
f(x
- t ) n + ( n t ) dt =
(IR
+ ( t ) dt)f(x),
E
L( T ) ,
x a.e. in
T.
Corollary 2.4 gives, in particular, a new proof of the result covered by Fejir's theorem, Theorem 4.2 in Chapter 11, but it does not enable us to identify the set of x's where convergence does occur. To see that this is the case, it suffices to note that an account of the estimates (2.8) and (2.9) of Chapter 11, K , ( t ) s c n 4 ( n t ) , where 4(t) = ,yFo,l)(-t) + t-2Xll,m)(t), 4 even. Because of the importance of this particular application, we state it separately as Proposition 2.5. Suppose f E L( T) and let a*(f)E wk-L( T) and
A}I
~ J { a * (> f)
(.*(A x ) = sup,la,,(f,
x ) l . Then
all A > 0.
s cllf((,,
Proof. Follows at once from Proposition 2.3. 3. THE CALDERON-ZYGMUND DECOMPOSITION
We may now return to the Calder6n-Zygmund decomposition of an integrable function f and express it in its most useful form. Theorem 3.1 (The Calderhn-Zygmund Decomposition at Level A). Let f E L ( G ) and let A > (1/27r) j T l f ( t ) l dt. Then there exists a sequence (4) of open, disjoint, dyadic subintervals of T such that If(x)l
s A, 1
AG -
141
I
I,
x
a.e. in
If(y)l dy
T\
s 2A,
uI j all j
(3.1) (3.2)
3. The Caldero’n-Zygmund Decomposition
and if 0,= R =
85
u 4, then
Moreover, if we set
(3.4) and
then f ( x ) = g ( x ) + b ( x ) ; the “good” function g and the “bad” function b are called the Calder6n-Zygmund decomposition off at level A and have the following properties lg(x)l s 2A, llgllp s
x a.e. in
(2~)P-’Ilflll,
(3.6)
T,
P
there, our conclusion
IV. The Basic Principles
86
4. THE MARCINKIEWICZ INTERPOLATION THEOREM
The Hardy-Littlewood maximal operator Mf maps L( T) into wk-L( T) and it also maps La( T ) into itself, with norm 1, since
What can we say about the behavior of Mf for Lp functions J; 1 < p < oo? Since these functions are in particular integrable, Mf is a well defined, wk-L function, but can we say something more precise about this function? It is convenient to cast this question in a general setting; first we present some definitions. An operation f + Tf is said to be sublinear if T ( f , +fz)and T(cfJ are well defined whenever Tfl and Tfzare defined, c is a scalar, and in this case
Such an operation T is said to be of (strong-) type ( p , p ) , 1 s p s co, if the mapping is defined in Lp(T) and for some constant c
II TfllP
c c Ilf
(4.3)
IlP
with c independent o f f ; the smallest constant in (4.3) above is the norm of T as a bounded mapping in Lp. Similarly, an operation T is said to be of weak-type ( p , p), 1 s p < 00, if the mapping is defined in Lp and for a constant c API{ITfl> h}l
S
cPIlfll;,
all h > 0
(4.4)
with c independent off; the smallest constant in (4.4) is called the weak-type norm of T as an operator of weak-type ( p , p ) . Since we will consider operations T naturally defined in a couple of spaces, it is convenient to introduce the sum and intersection of the Lebesgue spaces. More specifically, let LPo(T ) + Lpl(T) = {measurable, complexvalued functionsf: f = fo + fl,fo E Lpo(T), f,E Lpl(T)}, and introduce there the norm IlfllLPO+LPL
= inf{Ilfollp, + l l f l l I P l ~ ,
(4.5)
where the infimum is taken over all pairs fo E Lpo(T ) ,fl E LP1(T) such that f = fo + A ; then Lpoi LPl also becomes a Banach space. Similarly, in Lpo(T ) n LPl(T) we introduce the norm
IIfll LPOnLP'
= max(llfllm, IlfllPl)
(4.6)
4. The Marcinkiewicz Interpolation Theorem
87
and also Lpo n Lpl becomes a Banach space. Because only the linear structure of these spaces will be used at this time, we postpone discussing further properties until 7.3. We begin by proving a mixed weak-type-strong-type version of the Marcinkiewicz interpolation theorem; this is precisely what we need to handle the maximal function.
Theorem 4.1. Assume that a sublinear operator T is defined in LPo+ LP1 and is simultaneously of weak-type ( p o , p o ) with norm A l l dA,
(4.7)
[O,m)
which is readily obtained from the identity
by Tonelli’s theorem. In the first place, T is well defined in Lp, since, as is readily seen by considering large and small values of functions in Lp separately, Lp E Lpo + LP1.This consideration is not strictly necessary in our particular case as the underlying space is of finite measure and, consequently, Lp G Lpo; nevertheless, we prefer to give an argument that extends to the infinite measure case, such as the line R, with no modifications. Thus iff E Lp, we also have f = fo + f,, E Lp; i = 0, 1, and, by the sublinearity of T,
ITfWl =s l T f O W l + lTfdt)l ax. (4.8) first as this is the easiest case. From (4.8) it follows at
We assume p1 = 03 once that for A > 0,
mil’
{ITA > A ) {ITXOI’ A / 2 ) “ A m . (4.9) Moreover, since 11 T’II, S clJJfllloo, the second set in the right-hand side of (4.9) is empty provided that llfi llm
=s h/2c1.
(4.10)
This suggests that we actually consider a family of decompositions f = +f1,* parametrized by A. On account of (4.10) it is natural to set
f0,A
fl,A(t)
=
(s(‘)
( A / 2 c , )sgnf
if If(t)l S A/2c1, otherwise,
(4.11)
IV; The Basic Principles
88
and fo,*(t) = f ( r ) - f i , * ( t ) . By (4.9), (4.11) and the weak-type estimate, it follows that 1{1Tfl >
l{lTh,*>lh/2)1
(2/A)po@~~f0,A~~%
(4.13)
Whence, upon dividing by 2~ and taking pth roots, (4.13) reads
1I Tf (Ip
G
cPo/Pc;-PdP 4P1IP ( P -Po)l/p O
Il f
IIP.
(4.14)
But since p o / p = 1 - 7,1 - p o / p = 7 and p l / ps e"", 1 < p < co, we can rewrite (4.14) as
IITfll,
1
s c ( p -pO)(I-S)/po~~-~~:llfllp.
(4.15)
c S 4e1Ie independent off and T, and we are done in this case.
Next, suppose that p 1 < ax As usual we putf using (4.11) above we set
=
+ fi,*, but rather than (4.16)
4. The Marcinkiewicz Interpolation Theorem where E is a parameter yet to be chosen, at once that 2~11TfII;
C
P
I
AP-'I{l
89 =f-f1,*.
%,I,
From (4.8) we see
> A/2H dA
COP)
+P = I0
I
~P-'l~l~fl,hl
> A/2)1 dh
[O,oO)
+ 11,
(4.17)
say. The estimate for I. is carried out as before and it now reads (4.18)
To estimate I, we introduce the function (4.19)
Clearly, for t > 0
With this notation we have
whence, by integrating by parts, we readily see that
lo 00
Il = pAP-P14(A,A )
-
+(A, A ) dAp-p~ =J
+ K,
(4.20)
say. We estimate each term separately and do J first. For this purpose let J ( A ) = p A p - p l @ ( A , A). Since J ( 0 ) 3 0 the lower limit in J can be disregarded. As for J(co), we begin by observing that
IV. The Basic Principles
90
and consequently
If(t)Ipld t / A p l - p .
J s c lim sup A+m
(4.23)
I{,flSeA)
We now show that the right-hand side of (4.23) is 0. There are two cases, depending upon whether f is in Lpl or not. In the former case the numerator of (4.23) is a bounded function of A, whereas the denominator goes to 00 with A, and the limit is 0. On the other hand, i f f is not in Lpl, then the expression in (4.23) is indeterminate of the form co/co as A + co; this calls for L'Hopital's rule. The numerator of (4.23) can be written as
and the limit in question actually equals
But this expression is readily seen to be 0 since by Chebychev's inequality (EA)Pl{lf
>4 1
(4.24)
If(t)l"dt (lfl>&*)
and the right-hand side of (4.24) goes to 0 as A + 00 whenever f It remains to estimate K. On account of (4.21)
E
Lp.
(4.25)
Thus combining (4.17), (4.18), (4.20), and (4.25) we finally obtain
The time has come to select E, a prudent choice being that which makes both summands in (4.26) equal. This gives & =~ ~ ~ ~ ~ , - P ~ ~ q - (po))l/(P1-Po) c ~ p ~ / c ~ ( p
5. Extrapolation and the Zygmund L In L Class
91
and consequently the constant in the right-hand side above is 1 -
Po P L - P
1
CPP
( P- P o )
PI P - P o
P , - Po
1 pl-p -~
1 P-Po -~
PPI-PO
PPI-PO P1
P PI - P o C1
It is now a simple matter to verify that the constant c is as it should be. Because of its importance we emphasize the following particular case of the interpolation theorem. Proposition 4.2. Assume that 4 is an integrable function which satisfies the conditions of Proposition 2.3, and let F ( x ) = sup,lj,f(x - t ) n 4 ( n t ) dtl. Then C IlFllP
( p - l)l/P
Ilf IlP
1 A}l d max(hP, h 4 ) and the > A}[ d min(Ap, A").) norm in Lp + L" and jco.m~l{ljl 7.5 For 0 < a < 00, 0 < p < 1, the following statements are equiv .lent: (i) For every measurable set E E T, jE1f(x)l"dx s c@IP. (ii) f E wk-L"/"-B)( T). How is the constant c in (i) related to the weak norm o f f ? (Hint: (i) implies (ii) Put E = {lfl > A}. (ii) implies (i). Use
+
+
+
I,If(x)l"dx
6
I
lo,m)
min{lEl, I{x E E: If(x)l
'A}l} dA*.) up,,
Show by means of an example that Lln L strictly contains Lp. Show that condition (5.1) is actually equivalent to the single condition l{ITfl > 111 s c1 J[l/c2,m)l{lfl > t ) dt.
7.6 7.7
7. Notes; Further Results and Problems
103
a ( t )d t / t = 7.8 Let a ( t ) be a nondecreasing function such that O ( b ( s ) ) ,and let A ( t ) = jro,,) a ( s ) ds, B ( t ) = jro,t),b ( s ) ds. If T is a sublinear ) mapping simultaneously of weak-type ( 1 , l ) and type ( c o , ~ ~then I,A(ITf(x)l)dx s c + c I, B(If ( x ) l ) dx. (Hint: Theorem 5.3 corresponds to the case a ( t) = 1, b( t ) = In+ t.) 7.9 Let {g,,} be a sequence of wk-L functions with norm (uniformly) ~1 and furthermore let {c,,} be a sequence of positive numbers, C c,, = 1, and C c,,Iln c,l = K < 00. Then AI{C c,,g,, > A}[ S 2 ( K + 2 ) , all A > 0. (Hint: For each n > 1, put u , ( x ) = g,,(x) if g,,(x) < h / 2 and 0 otherwise, u,(x) = g,,(x) if g,,(x) > A/2cn and 0 otherwise, and w,,(x)= g , ( x ) ( u , ( x ) +-u.(x)). Put u ( x ) = C c,,u,,(x)and similarly for u ( x ) and w ( x ) .The following properties are readily verified: u ( x ) < A / 2 , I{u f O}l S C,l{gn > A/2cfl}l< 2 / A and
Finally estimate
This result is from Stein-N. Weiss [1969].) 7.10 Let { T , } be a sequence of sublinear operators defined in L ( T ) and assume that the weak-type ( 1 , l ) norm of these operators is (uniformly) S l . If { c is a sequence of positive real numbers so that c,, Iln c,, I < 00, then T = c,T, is of weak-type (1,l). 7.11 A sublinear operator T defined in Lp + L" is simultaneously of weak-type ( p , p ) and of type (00, co) if and only if there are constants cl, c2 such that for all f E Lp + L" and A > 0
.t:
1{1 Tfl > A l l s
I
6
[{lfl > s}IsP-l ds.
lc2A.m)
Extrapolate and compare with 7.7. 7.12 A sublinear operator T defined in Lp + Lmis simultaneously of types ( p , p ) and (00, co) if and only if there are constants cl, c, such that for all f E L p + L" and h > O
[
1{1Tfl > s}IsP-l ds s c1
[AS)
7.13 Show that for t large,
j
[a&m)
[{lfl > s}IsP-' ds.
IV. The Basic Principles
104
Extend the Marcinkiewicz theorem to the case where the operator T in question is simultaneously of types ( p o , p o ) and ( p l , pl). What can you say about the norm of T, a s a bounded mapping in Lp, po < p < pl? What if T is of type ( p o , p o ) ,weak-type (pl,pl) 1 < po < p1 < a? 7.15 State and prove the “abstract” version of the Marcinkiewicz interpolation theorem, i.e., T acts now on Lpo(X)+ L p ~ ( X where ), ( X , d p ) is an arbitrary measure space. Show that the requirement p o , p l , p > 1 is not necessary, i.e., we may actually assume p o , p l , p > 0. 7.16 There are ways to sharpen the conclusion of the Marcinkiewicz interpolation theorem by considering, for instance, a finer scale of intermediate spaces between the Lebesgue Lp classes. We may consider the Lorentz spaces Lps4(T ) consisting of those measurable f s such that
7.14
llfllp,q
=
(P
I
tos)
I{lA > A}14/PAq-1
dA)Il4 < co-
State and prove such a result. The work of O’Neil [1968] is relevant here. Can we use L In L as an endpoint in an interpolation theorem rather than as an intermediate space? Compare with 7.7 above. The results of Torchinsky [1976] are also of interest in this question. 7.18 A linear operator T defined in L + LP1, 1 < p1 < co, is said to be of pseudo-type ( 1 , l ) if to each f E L + LP1 and A > 0 there correspond a measurable set G c T and a function g E L( T ) such that [GIS cllflll/A, Ig(x>lc CAY llgll1 C l l f l l l and I T \ G I T ( f - g)(x)l dx S cllfIl1; all constants c above are independent off and A. Prove that if T is of pseudo-type ( 1 , l ) and of weak-type (pI,pl),then T is of weak-type ( 1 , l ) and consequently of type ( p , p ) for 1 < p < pl. (Hint: Estimate 1{1Tfl > 2A}l S 1{1T(f- g)l > A}/ + [{ITgl> A}l = 111 + IJI, say. IJI can be easily handled on account of the weak-type ( pl, pl) assumption. Also 111 G I GI + ((T \ G ) n { I T ( f - g)1 > A}/. This result is of interest in that it brings the CalderbnZygmund decomposition into play. We will have more to say about this later on; the proof is Cotlar’s (see Cotlar and Cignoli [1974]).) 7.19 A sublinear operator T is of weak-type (p, p) if and only if for each measurable subset E c T and 0 < r < p Kolmogorov’s inequality holds, namely, for all f E Lp(T ) 7.17
(Hint: Compare with 7.5.) Assume that T and S are sublinear operators and that T is majorized by S in the following sense: if C(x, r) = {y E T : r s Ix - yI =s2 r } , to each x in T and f~ L ( T ) there corresponds 0 < i < r such that ITf(x)lS inf,,,ccq,r)lSf(y)l.Then, if S is of weak-type (p, p) for some p > 0, so is T. 7.20
105
7. Notes; Further Results and Problems
(Hint: Let 0 < q < p. Then ITf(x)I" s infy,c(x,r)lSf(y)lq and averaging over C ( x , f )
where I ( x , T) denotes the interval centered at x of length 4T. The above estimate then gives ITf(x)I" s cM(ISFlq)(x),with c independent of x and f now; therefore, it will suffice to show that iVf((Sf(')(~)"~ is of weak-type ( p , p ) . But this is quite easy since by Proposition 5.1 ((M(lS''
> A'}I s cA-"
({S'
> t l / q } (dt
lcA4,W)
< CA-"
I
IlfllpPt-P/' dt
= CA - p
IIf 1;
;
[cAq,m)
the result is also Cotlar's.) 7.21 Assume that T and S are sublinear operators and that T is majorized by S in the following sense: to each x in T and f E L( T) there corresponds 0 < F < 7r such that if I ( x , f ) denotes the interval centered at x of length 2f, then
Then, if S is of weak-type ( p , p ) for some p > 0, so is T. (Hint: Use (7.1) now, this also is Cotlar's observation.) 7.22 With the notation of Section 6, we say that E G T is a set of divergence for B if there exists f E B whose Fourier series diverges at every point of E. Prove that E is a set of divergence for B if and only if there is f E B such that s*(f; x ) = 00 for x E E. (Hint: The condition is clearly sufficient; to show that it is necessary let g E B have a divergent Fourier series at each point of E and letf E B and { A j } be the function and sequence corresponding to g constructed in Proposition 4.6. Then, as in the proof of Proposition 6.3, we see that Is,(g, x ) - s,(g, x ) / s 2s*(f; x)A;>l and the Fourier series of g converges whenever s*(J x ) < 03. This observation is Katznelson's [ 19681.) 7.23 E is a set of divergence for B if and only if there exists a sequence of trigonometric polynomials { p n } such that IIpnllE1< 03
and
sup s*(pn,x ) = n
for x
E
E.
IV. The Basic Principles
106
(Hint: The proof is very much like that of Theorem 6.5. To show the necessity assume that the polynomials p,, of degree N,, exist and with integers k. such that k,, > knPl NnP1N,,, put f(x) = C,,eiknxp,,(x);then
+
smn+m(.L
+
X) - s m n - m - l ( . L
X) = eimnxsm(pn,
X)
whenever rn < N,, and the partial sums o f f diverge on E. Conversely, if E is a set of divergence for B there are a monotonic sequence p,, +. 00 and f E B such that Is,,(f,x)l > p,, infinitely often, for every x in E. Let now {A,,} be a sequence of integers such that - aAn(f)IIB < 2-" and choose integers 7, such that p,,. > 2 sup, s*(aAn,x). If we set now p,, = (2K2,,,+2* (f- aAn), then 1 IIpnIIB< co, and if x E E and n is an integer such that Is,,(f,x)l > p,,, then for some j,
[If
7j
< n < vj+l and Isn(Pj, x)l=
Isn(.L
x) - s n ( a * , ( f ) ,
x ) l > pn/2-
This proof is also from Katznelson's book [1968].) 7.24 Show that if the Ej's are sets of divergence for B, then so is E = uFlEj. 7.25 (Kolmogorov) Let {vk} be an Hadamard sequence, i.e., nk+l/n!, 3 8 > 1, and assume that f E L( T) has a lacunary series, i.e., f 1 c,,ke'nkx. Then show that s*(S, x) 6 ca*(f,x), where c depends only on 8, and limk+ms,,,(f, x) = f(x) a.e. (Hint: Observe in the first place that, if 1,cnk = s( C, l), then also C c,,, = s. This is not hard to see since we may suppose that s = 0, and in that case the partial sums s,,, of the series in question verify the following relation: if j > k, then (nj - nk)Snk = njan,-l - n p n k - 1 . Thus ( n j - nk)snk= o(nj) + o(&) = o(nj - nk), and Snk = o(1). Also Isnk[6 ( ( n j + nk)/(nj - nk))(T*d ((8 + i)/(8 - l))a*.) 7.26 (Kolmogorov) Let { nk} be a fixed Hadanard sequence and f E L ~T() . Then limk+ms,,,(.L x) = f ( x ) a.e., and, if n*(Jx) = supkIsn,(f, x>l, then Iln*(f)II,s cllfll, where c depends only on 8. (Hint: Both statements follow from the estimate
-
c
where c depends only on 8; but this is easy to obtain from an appropriate expression for s , ( f , x) - a,,(f,x) (cf. 5.4 of Chapter 11). The second assertion also requires the observation that suPIsnk(x)12 k
1Isnk(x) - ank(x)12+ u*(.L x)*) k
7.27 (Kolmogorov-Seliverstov) Iff E L2(T), then s , ( f , x) = o((1n n)'12) a.e. Furthermore, the function T*f(x) = ~up,,==~(1s,,(f, x)l/(ln n)'12) is in L2(T) and 11 T*fI12s cllfl12, with c independent of J: (Hint: It suffices to
7. Notes; Further Results and Problems
107
, ( ~ , n)’/’), with a prove the norm estimate for T % f ( x )= S U P , , ~ ~ ( I Sx)l/(ln constant independent of N. Let n ( x ) be any step function taking integer values 2 S n ( x ) d N, and put A ( x ) = l/ln n ( x ) ;it is enough to show that
I
=
I
A ( X ) S 2 , ( X ) ( f ;x
) dx S cllfll:
T
since T L f ( x ) = A(~)”’s,,(~,(f,x). By the estimates (1.15) and (1.17) in Chapter I, jTIDnCx)(x)I dx S c In N. Now I equals
&
jTA’”(x)sn(x)(f, x ) W )dx,
with
114112 = 1.
Put $ ( x ) = A’/’(x)+(x).Then
say. Clearly
and the estimate obtains using the above bound for jTIDnCx,(x)I dx. This shows that s,(f, x ) = O((1n n)”’) a.e. To refine 0 to o we pass to lim sup instead.) 7.28 (Plessner) I f f E L( T) and
is in L2(T), then the Fourier series off converges a.e. (Hint: Show that the assumption is actually equivalent to ~,n,,l~cn(f)~’ In n < co.) 7.29 With the same notation and assumptions of 7.14 of Chapter 111, show that if
IV. The Basic Principles
108
and 8 such that
then there is a constant c, depending only on
Furthermore IIs*I12 6 cllfl12 and sN(x)+ f ( x ) a.e. (Hint:The proof is an interesting application of an “averaging” method. Let I = [x, x + 27r/ tN]. If y E I, then N
lsN(x) - sN(Y)I
Irk1 If,(@)
-fk(fky)l
k=O
and consequently lsN(x)I S IsN(y)l+ c(CIrk12)1’2.By averaging now over I, it follows that
say, where RN+,(y)in B is the “tail” of the series. Clearly, C is of the right order and A S c M f ( x ) .To do B change variables y = x + s/ tN, 0 S s S 21r, and apply 7.14 of Chapter 111 to RN+I(x + S/tN) = C:=”=,+, rkfk(x+ S/tN). ~ follows at once. Thus the first assertion is proved; that I I s * ~S~ c(Clrk12)1’2 To prove the a.e. convergence apply the norm result to R:(X) = SUpj3ksN\Sj(X)- sk(x)I; since we have ro = * * = rN = 0, 11~511~ + o as N + 00. Finally, note that since Rf 3 3 R53 2 0 and R & ( x ) dx + 0 we also have limN+mRK(x) = 0 a.e. This result is also from Meyer’s work [1979].) 7.30 Suppose {S,}ueIis a family of transformations from T into T verifying the following properties: (i) each S, is measure preserving, i.e., if E is a measurable subset of T, then [S;’(E)l = IEl, and (ii) the family is mixing, i.e., if El and E2 are measurable subsets of T and r > 1, then there is an S, in the family such that IE, n S;’(E2)1 S rlE,(IE21.(Observe that if the S,’s were such that IE, n S,’(E2)l = IElllE2),then E l and S,’(E2) would be probabilistically independent; our assumption is not so restrictive.) Prove that if { E k }is a sequence of measurable subsets of T such that IlEkl = 00, then there exists a sequence {S,} of S,’s such that almost every x in T is in infinitely many of the sets S;’(Ek). 7.31 (Sawyer) Suppose that { Tk}is a sequence of linear operators defined in Lp(T) for some 1 s p =S 00 which verifies the following assumptions: (i) each T k is continuous in measure, (ii) each Tkis positive, i.e., iff 3 0, then Tkf 3 0, (iii) there is a family {S,},EI verifying the assumptions of 7.30 so that the Tk’sand Su’s commute in the following sense: i f f € L p ( T )and
I,
-
-
-
7. Notes; Further Results and Problems
109
( S , f ) ( x ) = f ( S , x ) , then for each Tk, TkS, = S,Tk; more precisely, for each x E T, T k ( S , f ) ( x ) = ( S , ( T k f ) ( x ) . Prove that for such a sequence of operators the following conditions are equivalent: (a) supkl Tkf(x)I = T * f ( x ) is of weak-type ( p , p) and (b) for each f E Lp(T ) , T * f ( x ) < 00 a.e. (Hint: As in the proof of Calderh’s theorem, assume that T* is not of weak-type ( p , p) and reach a contradiction; 7.30 is an important step in the proof which follows de Guzmh’s presentation [ 19811. Sawyer’s principle has wider applications than it may seen on the surface; in fact, although the T k ’ s themselves may not be positive, they may be of the form Tk = PkK where the Pk’s are positive and T is some fixed, bounded, linear operator in Lp. Then the estimate for T* will automatically follow from the one for P*.) 7.32 (Baire Category Theorem) A metric space is said to be of first category if it can be written as a countable union of sets that are nowhere dense. If a metric space is not of first category, then we say it is of second category. The rational numbers on the line with the usual metric are of first category; on the other hand, the real line is of second category. This last assertion is a special case of the Baire category theorem: A complete metric space X is of second category. (Hint: Suppose X = UGl Xi,where the Xjare nowhere dense, that is, the sets have no interior points. Fix a ball B ( x o , 1); since XI does not contain it, there is a point x1 E B(x,,, l ) \ x l ; from this point on the proof is identical to that of the uniform boundedness principle, Theorem 2.3 of Chapter I.)
xj
CHAPTER
V The Hilbert Transform and Multipliers
1. EXISTENCE OF THE HILBERT TRANSFORM
OF INTEGRABLE FUNCIIONS This section covers one of the basic results in harmonic analysis, namely, the (a.e.) existence of the principal value integral defining the Hilbert transform of a summable function. This limit will not, in general, be a locally integrable function, though. The proof we present here is purely real variable and it serves as an inspiration for the extension of these results to the Euclidean n-dimensional case.
Theorem 1.1. Suppose that f E L ( T ) . Then
'I
lim HEf(x) = lim -S+O
E+O
7r
f(x - 2 ) 2 tan(t/2) dt
the Hilbert transform exists for almost every x in T. We call this limit Hf(x), off, and the relation (1.1) is denoted by
Hf(x)
= p.v. - -
dt.
The principal value notation (P.v.) above emphasizes that a symmetric neighborhood about the origin is deleted before the limit is taken; the origin corresponds to the singularity of the convolution kernel here. Proof. The idea of the proof is to show that the p.v. integral converges a.e. in the complement of a sequence of sets with measure approaching 0; once this is achieved, clearly the p.v. integral exists also in the union of these sets, namely, in a subset of T of full measure. 110
111
1. Hilbert Transform of Integrable Functions
In first place we may, and do, assume that f is nonnegative. Let A be a the average of I over T, will do. We then invoke large constant; A > the Calderbn-Zygmund decomposition o f f at level A, Theorem 3.1 of Chapter IVYand obtain a sequence of open, disjoint subintervals ( 4 ) of T and a decomposition f = g b, induced by these intervals. Clearly, for each E > 0 H , f ( x ) = H,g(x) H,b(x), x in T. Moreover, since g E L 2 ( T ) ,by Corollary 6.5 in Chapter 111, lim,,o H,g(x) = g ( x ) exists a.e. in T. Let, as usual, R* = U2 4 , IR*ls(2/A) j T l f ( y ) l dy. Suppose we can show that
If
If IT,
+
+
lim H,b(x) E+O
exists a.e. in
( 1-21
T\R*.
Then by choosing a sequence Ak + a,and if {RE} denotes the corresponding sequence of open sets, the p,v. integral in (1.2) exists, for the corresponding b’s of course, a.e. in the complement of R z and the same is true for the p.v. integral corresponding to J: Thus the Hilbert transform off exists T\R?), which is a subset of T of full measure. a.e. in Uk( Returning to the proof of (1.2), then, it suffices to show that { H , b ( x ) } is Cauchy a.e. in T\R* as E + 0. In other words, we must show that lim \H,b(x) - H,b(x)J= 0,
a.e. for x in
T\O*.
(1.3)
&,V+O
We break the proof of (1.3) into two steps. In the first place, we show that lim suplH,b(x) - H,,b(x)l < a,
a.e. for x in
T\R*,
(1.4)
E,11+0
and once we know that this lim sup is finite we show, in the second step, that it is 0. For E > 7 > 0 fixed, note that H,b(x) - H,b(x) equals
Moreover, since b ( t ) = C j b(t),yIj(t)and x f? 4 for any j , we can divide the 4’s into 2 families, namely, those intersecting [ x - E, x - 7 ) and those intersecting ( x + 7, x + E ] ; all other intervals contribute nothing to the integral in (1.5) and may be disregarded. As the argument for each family is identical we only consider those intervals 4 intersecting ( x + 7, x E ] . These intervals can in turn be separated into three subfamilies, to wit
+
(i) Possibly an 4 containing x + 7 , call it 1;. (ii) Possibly an 4 containing x + E, call it 1;. (iii) Those I’s totally contained in ( x + 7 , x + E ] , call them still 4.
112
V . The Hilbert Transform and Multipliers
The main contribution to (1.5) comes from the integrals extended over the 4’s; the other terms represent basically “error” terms. We make this statement precise. Let L; = length of 17, x; = center of 1; and put
The following two geometric observations are clear: (a) (b)
Since x E 21;, then r] > L;/2. ( x 7,x + E ] n 1; c ( x + 7 , x
+
+ 7 + LT) G ( x + 7,x + 377).
Thus
IA7l zs
I
Ib(‘)l
(x+1),x+31))
-
dt.
(1.6)
2ltan((x - t)/2)1
-
Moreover since tan u 77 when u 7 and 7 is small, the denominator of the integrand in (1.6) is of order c / 7 and it follows that Ib(x + t)l di
Ib(t)l dt = 77 C
Ib(x+t)ldt.
s-j
77
(1),3r))
(1.7)
(031))
Now b ( x ) = 0 since x E T\a* and (1.7) can finally be rewritten as
The expression appearing in the right-hand side of (1.8) is a familiar one, as we have previously encountered it in Proposition 4.1 of Chapter 11; it actually is o ( 1 ) as 7 + 0 at every Lebesgue point x of b that is a.e. in T. Similarly, C
c3E
J
Ib(x + t ) - b(x)l dt
= o(1)
(0.3~)
also a.e. in T. So these are indeed error terms.
as
E
+O
(1.9)
1. Hilbert Transform of Integrable Functions
113
To estimate the main contribution we invoke the cancellation property = center of 4
jI, b(t) dt = 0 and rewrite with xi
(1.10) say. Furthermore, a simple computation making use of elementary trigonometric identities gives k(x, t, x j ) =
sin( ( t - xj)/2) sin((x - t)/2) sin((x - xj)/2)
(1.11)
and to estimate k(x, t, xi) we refer once again to the geometry of the situation. In the first place, notice that if t, xj E 4 and x E 24, then (1.12) Ij I
I
,
t
\
I
x.
J
21j 1
,
X
Indeed, since with Lj = length of 4 we have Ix - xjl > Lj/2, then we see at once that Ix - tl Ix - xi[ + Ixj - tl G Ix - xjl + Lj s 31x - xi[. Exchanging the role of xi and t we also note that Ix - xjl s 31x - tl, and combining these two estimates (1.12) follows. Moreover, since we are dealing with small values in the above expressions, we may replace the sines there by their arguments and on account of (1.12) (and estimate (1.16) of Chapter I) we observe that (1.13) This is all we need to complete the estimate of the Aj's. Indeed, by (1.10) and (1.13) it follows that
V. The Hilbert Transform and Multipliers
114
with c independent of x and j . Combining these bounds with the ones obtained for the interval [ x - E, x - 7 )we finally see that for almost every x in T\R*,
I H A x ) - H,b(x)l
=
L&,,(X),
(1.14)
say. Thus
say. In view of (1.15) estimate (1.4) will hold once we show that L ( x ) < 00 a.e. For this purpose we intoduce a majorant of L ( x ) , the Marcinkiewicz function A(J x) associated to R = I j and given by
U
(1.16)
We claim that
A(J x ) < 00
a.e. in
T\R*.
(1.17)
The usual procedure for this is to note that the stronger statement r (1.18) I = J A(Jx)dx 0, we may always miss the first N intervals in the family (4) as soon as E , 77 > 0 are small enough in (1.14). Furthermore, since the right-hand side of (1.21) is the tail of a convergent series for almost every x in T\R* we obtain that L(x) = 0 for those x's by letting N + 00 in (1.21); this means that (1.3) holds. Remark 1.2. We isolate two facts from the proof above. First, by putting E = m in (1.14) we see that for almost every x in T\R*
IH,b(x)l
'I
W f x) ; +77
Ib(t)l dt +
(X+,,X+3,)
C ( A U x) + Mb(x)),
77
I
Ib(t)l dt
(x-3,T-V)
( 1.22)
71 > 0
with c independent of 7. Moreover, once again by (1.14) but taking now limits as 77 + 0, we note that for almost every x in T\R* (1.23)
IWX)l zs c A(f, x). 2. THE HILBERT TRANSFORM IN Lp(T),1
0
hl{lH&fl> A l l c C l l f l l l .
(2.2) Proof. We may assume that A > llflll; we then invoke the Calder6nZygmund decomposition at level A and write, in the notation of Theorem 1.1, f = g b. Also for x E T\R*,
+
H&f(X)
= H,g(x)
+ H&b(X).
(2.3)
Moreover, since {IHJ(x)l>
A}
c Q* u {x E T\Q* :IHef(x)I > A},
(2.4)
it suffices to estimate the measure of each set on the right-hand side of (2.4). In first place
IQ*l
2 C l l i l c 4Tllflll/A and this bound is of the right order. Also by (2.3) {x
E
(2.5)
T\R*: IH&f(x)l > A }
c {x E T\R*: IH,g(x)l> h/2} u {x E T\R*:
IH,b(x)l> h/2)
=IuJ,
say. To estimate Z we make use of the L2 result for H,g. From Chebychev’s inequality, estimate (6.2) in Chapter I11 and estimate (3.7) in Chapter IV,
which in turn implies that 1 1 1s C l l f l l l / A ,
and this term is also of the right order. As for the set J, by estimate (1.22) of Remark 1.2, we note that it is contained in {x
E
T\R*: A(J x) > A/4} u {x
E
T\Q*: M b ( x ) > A/4}
= J1 u Jz,
say. By estimate (1.20), lJll
s Cllflll/A.
(2.7)
2. The Hilbert Transform in Lp(T), 1 ==p < 00
117
Also by the maximal theorem and estimate (3.9) of Chapter IV we obtain cIIbIIi/A C I I f I I i / A (2.8) Thus, by combining (2.5)-(2.8), and since all constants are independent of E and f; the desired conclusion follows. IJ2I
Observe that actually H , f ( x ) , being the convolution of integrable funcH,flll = 00 in tions, is also integrable for each E > 0; nevertheless, limE+.o~~ general. A reasoning essentially analogous to that of Theorem 2.1 obtains Theorem 2.2 (Kolmogorov). Let f~ L ( T ) . Then there is a constant c, independent off; such that for all A > 0
> All c Cllflll.
(2.9) Proof. We use estimate (1.23) of Remark 1.2 in this case; the proof is simpler than that of Theorem 2.1 since the term J2 does not appear. AI{IHfI
It is now an easy task to complete the consideration of the Lp result as well. We do the case 1 < p < 2 first, which in view of the L2and weak-type (1,l) estimates we expect to be rather straightforward. Theorem 2.3. Suppose f E Lp(T ) , 1 < p < 2. Then there is a constant cp = O(1/( p - l)), independent of E and f; such that llH€fllP cpllfll,.
(2.10)
Theorem 2.4 (M. Riesz). Suppose f E Lp(T), 1 < p < 2. Then there is a constant cp = O(l/(p - l)), independent off; such that
IIHfllP
(2.11)
CPllfllP.
Moreover, H f ( x ) = f ( x ) a.e. and consequently Lp(T) admits conjugation. Proof of Theorem 2.4. Let l / p = (1 - 7)+ 7/2,0 < 7 < 1; then by Theorem 4.1 in Chapter IV (with po = 1, p, = 2 there) H is an Lp bounded mapping with norm O( l / ( p - 1)('-')). Furthermore, since ( p - 1)' d 1 and actually tends to 1 as p + 1, cp = O(l/(p - 1)) as anticipated. In order to show that H f ( x ) coincides with f ( x ) a.e., we consider H ( f - a , ( f ) ) ( x ) , which equals H f ( x ) - c?,,(f; x ) a.e. since o n ( fE>L 2 ( T ) . Now by (2.11)
IIW-
Gn(f>IIp
cpllf-
an(f)Ilp,
(2.12)
with cp independent of n. But the right-hand side of (2.12) goes to 0 as n + 00 by FejCr's Theorem 2.7 of Chapter 11, and so does the left-hand side. Thus, on account of Proposition 1.1 of Chapter I,
c j ( H f )= limn+.m cj(c?"(f)), allj.
K The Hilbert Transform and Multipliers
118
Moreover, since c j ( G n ( f ) >= (1 - Ijl/(n + l))(-i)(sgnj)cj(f) when Ijl d n and 0 otherwise, we see at once that the above limit is (-i)(sgn j)cj(f); this implies that f = Hf E Lp(T). H The case 2 < p S 00 remains to be discussed. When p = 00 we may, and will, give more than one answer; this is because f E L“(T) does not, in general, imply that Hf E La( T), as the simple example in Remark 2.7 below shows. Before discussing this negative result, we consider the positive results for 2 < p < 00. They are readily obtained by a “duality” argument.
Theorem 2.5. Suppose f E Lp(T), 2 < p < 00. Then there is a constant cp = . O(p), independent of E and f; such that IIHEfllP 6 CPllfIlP.
(2.13)
Theorem 2.6. Suppose f E Lp(T), 2 < p < 00. Then there is a constant cp = O ( p ) ,independent off; such that llHf
IlP
s
CPIlfIlP.
(2.14)
Moreover, Hf(x) = f ( x ) a.e. and consequently Lp(T) admits conjugation.
Proof of Theorem 2.6. Since Lp c L2, we already have Hf = f E L2(T) by the results in Section 6 of Chapter 111; it only remains to prove that Hf E Lp(T) as well. By Theorem 3.3 of Chapter I1 it suffices to show that IIantf)IIp
s
CpIIfIIp
(2.15)
with cp = O(p) independent of n and f ; this is not hard to do. Indeed, let g be a trigonometric polynomial with llgllp. d 1, l/p + l/p’ = 1. Then by Parseval’s identity (iv) of Theorem 1.8 in Chapter 111 it follows that
Whence by Holder’s inequality, estimate (3.4) of Chapter 11, and Theorem 2.4 we obtain
III
cp*llfllp* (2.15) follows now by the converse to Holder’s inequality but with cp,instead of cpthere. This is only a minor inconveniencebecause since p = pf/(p’ - l), pf near 1 and c,. = O( 1/( p’ - l)), then also cp,= cP = O(p). l l ~ n ~ ~ ~ l l p ~ l l f l l p
Remark 2.7. The order of the constant cp in Theorem 2.7 is sharp. Indeed,
2. The Hilbert Transform in Lp(T), 1 < p < w let f ( t ) = x[o,ll(t). Then clearly c(ln(sin(x/2)1- lnlsin((1 - x)/2)l), and
= c(
I[,.,
l f 11,
sPe-’ ds)
119
= 1,
all
-=. r ( p + c
p
3
i)lIP.
1,
Hf(x)
=
(2.16)
Here r ( p + 1) denotes the gamma function of order p and by known estimates from (2.16) we see at once that lim infp+mllHfllp/p3 c > 0. Note in passing that, although Hf is not bounded, Hf E Lp,Hfis exponentially integrable, and IHf(x)l has logarithmic growth. These properties remain true for the Hilbert transform of every L“ function, as we will prove shortly (however, the last property will have to wait until Chapter VIII). Since Lp(T) admits conjugation for 1 < p < co, IlS,,(fl -flip + 0 as n + 00. We also have in this case
np<m
Proposition 2.8. Assume that f E L P ( T ) ,1 < p < co.Then as n + co. Proof. Since ;,(A n+m.
-f
=
(sn(n-n-,
IIfn(f)
To complete the discussion of the case p
--711p =
IIs”(f) -flip + 0
CpIISn(f)
-fIIp+=
0 as
1 we show
Proposition 2.9. Assume that f E L( T ) and 0 < p < 1. Then IIHfll, c p ~ ~ cp f ~=~0(1/1 l , - p ) independent off: A similar estimate holds for HE with cp also independent of E.
Proof. Since Hf E wk-L( T) we are in a position to use 7.5 of Chapter IV; we put (Y = p and /3 = 1 - p there and note that the constant is of the right order. W One of the first results we discovered was that 11 H E f - HfII, + 0 as E + 0. It is reasonable to expect that a similar result will hold for Lp norms, 1 < p < 00, as well as substitute results for p = 1; we prove this next. Proposition 2.10. Assume that f E Lp(T ) , 1 < p < a.Then limll HJ - Hfll, E+O
= 0.
(2.17)
Proof. For an arbitrary f~ L P ( T )let g be a trigonometric polynomial so that Ilf- gllp < 7 is arbitrarily small. Then by Minkowski’s inequality
V . The Hilbert Transform and Multipliers
120
we have
IIH€f-H f l l P
s
I I H E f - H&llP + IIKg - HgII,
+
llm?-
s c p l l f - gllp + IIKg - Hgllp + cpllf-
WllP
SllP
s 2crl+ IIH&g- HgIIp. - HgII, = 0. But since g Consequently, it suffices to show that limE+ol[H&g is a trigonometric polynomial, the proof for estimate (6.4) of Proposition 6.2 in Chapter I11 also works in this case.
Proposition 2.11. Assume that f E L( T ) . Then
lim H E f = Hf;
in measure,
E-0
(2.18)
and - Hfll, = 0, limllHEf €’O
0 < p < 1.
Proof. The convergence in the Lp metric follows as above and (2.18) is an immediate consequence of this. W
The truncated Hilbert transforms, as well as the Hilbert transform itself, are uniformly controlled by the “maximal Hilbert transform H*” given by H * f ( x ) = SUP IH€f(X)l.
(2.19)
O A } [ S cllfll,,
all A > 0.
(2.21)
Proof. As in the proof of the weak type for Hf itself we invoke the Calder6n-Zygmund decomposition at level A sufficiently large, and write f=g+b,R*=U21,..Then H * f ( x )< H * g ( x )+ H * b ( x )
and it suffices to show that Al{H*g
’
S cllfll1
(2.22)
(2.23)
and Al{x E T\R*: H * b ( x ) > A/2}1 S cllflll.
(2.24)
This is straightforward. Indeed, by Theorem 2.13 11 H*gll, S cllgll, S CA 11f [I1 and (2.23) follows from Chebychev’s inequality. Moreover, by estimate (1.22) in Remark 1.2 we note that H * b ( x ) S c ( A f ( x )+ M b ( x ) ) ,x E T\R* and (2.24) follows at once on account of estimate (1.2), the fact that the maximal operator is of weak-type (1, l), and )Ib 11 S c 1) f 11 Remark 2.14. We may now combine Theorem 2.13 with the Marcinkiewicz interpolation theorem to find that, in Theorem 2.12, cp = O ( l / ( p - 1)); thus the analogy with the Hilbert transform itself is complete.
3. LIMITING RESULTS Assume f E L( T ) ,f
- 1 cje? Still the question remains as to when 1(-i )(sgn j )cje (3.1) gx
V. The Hilbert Transform and Multipliers
122
is a Fourier series. Suppose Hf E L ( T ) . Then since Hf = f * p.v.(l/(tan(t/2))) (in D') by 7.22 in Chapter I11 we get that c j ( H f ) = (-i)(sgnj)cj and (3.1) is a Fourier series in this case. To decide when Hf is integrable we proceed as we did for the maximal operator and search for an extrapolation result first. Proposition 3.1. A sublinear mapping T defined in L + Lp, 1 < p < 00, is simultaneously of weak-types (1,l) and ( p , p ) if and only if there are constants c, , c,, c3 such that for all f E L Lp and A > 0
+
14 Tf I > A l l
I,,
CI(+
s"-'l{lf
I > s)l
ds
.c2A )
+
A
[
I{lfI > 41 d s ) .
(3.2)
[s*,00)
Proof. The necessity follows from the decomposition (4.14) of Chapter IV with E = 1 there. As for the sufficiency, if (3.2) holds and f E Lp say, then in the first integral we just replace c2A by 00 in the limit of integration, and this term is of the right order. It is also readily seen that the second integral does not exceed
which is also of the right order. The weak-type (1,l) case is handled similarly. Finally note that (3.2) is equivalent to the single condition obtained by setting A = 1 there. We can now extrapolate. Proposition 3.2. Assume T is a sublinear operation defined in L + Lp, 1 < p < co, which is simultaneously of weak-types (1,l) and ( p , p ) . Then T maps L In L into L and there are constants A, B independent off such that
11 Tflll
A
+B
IT
If(t)lln+lf(t)l dt.
(3.3)
Proof. It sufficesto show that ~~1~,1,1~~ Tf (t)l dt is bounded by the right-hand side of (3.3). But this is immediate since by (3.2) it suffices to show that the same is true for
(3.4)
3. Limiting Results
123
and (3.5) That the expression in (3.5) is of the right order was already proved in Theorem 5.3 of Chapter IV; the expression in (3.4) is quite easy to handle. Indeed, suppose as we may, that c2 Q 1 and observe that (3.4) is dominated by 1 sP-l ds dA
I,,,,, P I,, ,
I,, +LC2*)
sp-'({lfl > s}l ds dA
+
A+ sA
J
~P-'IWI > S)I
1 dA ds
[S/C2.-3)
IC2.W)
+B J
[
){If1
AP
> s}) ds.
lc2,m)
Proposition 3.2 clearly applies to H E , H, and H*. There also is some evidence, furnished by estimate (2.1), that, as in the case of the maximal operator, some kind of converse result may be true. Indeed, for positive, integrable functions f vanishing off [0, m / 2 ) with Hf E L( T), we have that
*>
IHf(x)l dx
3 c
[
(O,m/21
lro,xlf(r) dt dx
f(t)
= C! [O.r/21
=c
x
I
1 -dxdt sr/21 X
f ( t ) ln(m/2t) dt,
CO,r/21
which is readily seen to imply the integrability of f ( t ) ln+f(t) in [ 0 , r / 2 ) . The positivity o f f is important here. We shall return to this question in Chapter VII and remove the restriction on the support off then. There is yet another extrapolation result and it corresponds to the spaces near L", since, unlike the maximal operator, the Hilbert transform is not bounded in La( T). We may search for this result by a duality argument as follows: for trigonometric polynomials f and g observe that
V. The Hilbert Transform and Multipliers
124
On account of Young's inequality (see 5.2 of Chapter I), we expect now some kind of exponential integrability for f when f~ L". The precise statement for general operators T that behave like the Hilbert transform is
Theorem 3.3. Assume T is a sublinear operator bounded in Lp,1 < po s p < 00, with norm O ( p ) as p + 00. Then there exist constants c and A independent o f x such that (3.6)
Proof. We estimate the integral in (3.6) by
c
+c
" 1 1 G A k ( k+ l)kl(fll&, kapo
which is readily seen to be dominated by an absolute constant provided A is sufficiently small, since ( k / e ) k k ! . H This general theorem indeed applies to H E , H, and H*, but is there a more precise result for the particular case of the Hilbert transform? In other words, can we be more explicit about the value of A in the conclusion of Theorem 3.3? In order to do this and to illustrate the simplicity and power of the so-called complex method, we prove the following result Proposition 3.4 ~ / 2 Then .
(Zygmund). Assume f is a real-valued function, If(x)l s evCx)lc o s ( f ( x ) ) dx < 2 cos (;PIT
f(t) dt
)
s 2.
Proof. Suppose first that f is a trigonometric polynomial, f ( x ) = and put F ( x ) = i ( f ( x )- i f ( x ) ) .F ( x ) has the following properties
c,:
(3.7)
qe""
cje""). (i) It equals i(co+ 2 (ii) If F ( z ) = i(co 2 C,:, cjzJ),then F ( z ) is a harmonic function in Iz( s 1 (actually it is analytic there). (iii) F ( 0 ) = ico = i((1/27r) j T f ( f ) d t ) , (co(s 7 ~ 1 2 . ) also harmonic and analytic in IzI s 1. (iv) e F ( =is By the mean value property of harmonic functions, to be proved in Chapter VII, we have
+
3. Limiting Results
125
Whence replacing F ( e k ) by its explicit form and taking real parts in (3.8) we see that
e R x )cos(f(x)) dx
= Re =
((1/27r)
ef(x)e*(x) dx JT
-
Re( eF(’))= cos( c,).
(3.9)
Also, replacing f by -f in (3.9), we see that
& j, ,-m
cos(f(x)) dx
(3.10)
= cos(c,)
and adding (3.9) and (3.10) we get
kI,
(3.11)
eliCx)lcos(f(x)) dx 6 2 cos(c,).
Next we would like to show that (3.11) still holds for an arbitrary, real-valued function f with If(x)l s 7r/2. To see this, consider the FejCr polynomials a,,(f;x) = f * 2K,,(x); a,,(f,x) is real valued, Ila,,(f )llm ( ~ / ~ ) I I W , I=I I m/2, and c,(a,,{f)) = c,, all n. Furthermore, since Ila,,(f)-fl12 + 0, then also IlG,,(f) -fllz + 0 and there is a subsequence nk -+ co such that a,,,(f;x) +f(x) and G,,,(f; x) + f(x) a.e. in T. Since (3.11) holds with f replaced by u,,,(f)there and the right-hand side is independent of nk, by Fatou’s lemma we see that (3.11) holds for f as well. Corollary 3.5 (Zygmund). Suppose f is real valued and dx < co. Then
I,
Proof. Let
llfllm
=
~
~
A}, BA = { f < -A}. Then 5.11
I,
AIA,I < j A A j ( x dx ) = - j , f ( . ) ~ , ( x ) dx =Z CII flip'
repeat the argument for BA.)
11x4
Ilp
=
c l l f l l p ‘ lAAll’p;
135
5. Notes; Further Results and Problems
-
There is a bounded function f(x) (sin nx)/n = ( T - x)/2 whose conjugate f(x) (cos nx)/n = -lnlsin(x/2)1 is unbounded. 5.15 There is a continuous function f(x) = C:+(sin nx)/(n In n) whose conjugate f(x) c:=,(cos nx)/(n In n) has a Fourier series which converges everywhere except at 0. (Hint: To show the continuity invoke 7.24 in Chapter 111: actually, the partial sums off diverge to 00 at 0, which means that the series also diverges (C, 1) to co there.) 5.16 Assume J; f E L“( T) and that in addition ~ ~ s n ( f ) ~S~ m A, all n. Then s B. there is a constant B such that also Ils’,,(f)llDO 5.17 Assume now that J; f~ C ( T ) and limn-rwllsn(f)- f l l w = 0; then limn+wll.Sn(A -f1lw = 0. 5.18 Assume f E C ( T ) , 1lfllrn s 1. Then there are constants cl, c, independent of n so that I{Isn(f)l > A}l s qe-4”. 5.19 Suppose f E L In L and show that IIs, , 5.14
- c:=’=,
-
and
5.20 Parseval’s relation holds for f _E L In L( T) and g E Lm(T). 5.21 Assume f E C(T). Then eAlf(x)l dx < 00 for each A > 0. Show that as p + m. (Hint: Write f = f - g g, where g is a IIfII, = o ( p )
I,
+
trigonometric polynomial with Il f - gllm sufficiently small.) 5.22 There is an absolute constant A. > 0 such that if l l f l l w G 1, then for 0 < A S Ao, I, eAISn(f.X)I &9 T eA15m(f.x)I h < c, where c depends only on A. Iff E C ( T), the estimate holds for each A > 0. (Hint: Consider, as usual, the modified partial sums s’f(f; x) off given by
and observe that rewriting sin n t = sin n( t + x) cos nx - cos n( t + x) sin nx it follows that s’f(J; x) = &(x) sin nx - &,(x) cos nx, where gn(x)and h,(x) aref(x) cos nx andf(x) sin nx, respectively, Ig,,(x)l, Ih.(x)l 4 1. Therefore,
5.23
Assume f is as in 5.22. Then there is an absolute constant A. > 0, so
V . The Hilbert Transform and Multipliers
136
l / p + l / p ' = 1 . If A is sufficiently small, and p < 00 sufficiently large so that Ap' is still small, the integral above is bounded and the term IIsn(f) - f l i p = o(l).) 5.24 If elf(x)'"dx < 00, then eAli(x)Isdx < 00 provided /3 = a / ( a + 1) and A is sufficiently small (Hint: A way to do this is by dualizing the result If(x)l(ln+lf ( x ) l ) a dx < 00 implies If(x)l(ln(2 + I ] ( X ) [ ) ) ~ - ' dx < 00, A > 0. Of course, there is always a direct proof.) 5.25 Let A ( u ) be a continuous, nondecreasing function, A ( 0 ) = 0, such that lim sup,,m A ( 2 u ) / A ( u )= co. Then there exists a function f so that A(lf(x)l) dx < 00, yet A ( l j ( x ) l )dx = 00. (Hint: Construct by induction an increasing sequence of positive numbers uk so that A ( 2 u k )> 2 2 k A ( ~> k )2, k 3 1, and uk 2 2 4 - 1 , k 3 2. Next define positive integers 1 s n, < n2 < * * so that 2-k-' < 2-"kA(uk)s 2-k, k 3 1 , and define the function x E [-2-"k, -2-nk+l ) k = 1,2, ..., x E T\[2-"1,0). =
IT
I,
IT
IT
I,
I,
-
{ :,"'
It is readily seen that
IT A(f ( x ) ) dx
0, A > 0, let E = { x E T :lf(x)l > AT, M f ( x ) s 7 ) . Then IEl S ~ ~ ~ f ~ ~ where ~ e - c~is' independent ~ / q , off, A and 7. (Hint: We may assume that 7 > llflll, for otherwise E is empty, and that A 2 2, for otherwise the result follows from the weak-type estimate for .f We may also assume that Mf( P ) s 7 and write the open set { Mf > 417) = u4,where the Ts are disjoint open intervals, and put
and b=f-g. Note that M f ( x ) > 7 for x E U 3 4 , and, consequently, E s E , u E 2 , where El = {lg"l > A7/2} and E2 = { x E T\U34: 16(x)1> h17/2}. To show that lEll 6 cllfll le-A/c/7we use 5.27, and to estimate lE21 we use 5.26; this proof is also Muckenhoupt's and extends a result of Hunt [1972].) 5.29 The most general multiplier is a sequence { A j } with the property that for each f E C"( T ) , f cjeVx, Ajcje""E D', we denote this class (C", D'). Show that the characterization of (C", D') is also as general as possible, namely, { A j } E (C", D') if and only if the sequence { A j } is tempered. 5.30 Another natural question is how to characterize (C, M ) , i.e., to identify those sequences { A j } which transform continuous functions into finite measures. This was done by Gaudry [1966] as follows: let B = { h e C ( T ) : h=C,:,lf;*gj, CIIlf;IImllgjll, r ] } is empty for r ] sufficiently large.)
+ I',"=,
-
I:=,
(I:=,
CHAPTER
Paley's Theorem and Fractional Integration
1. PALEY'S THEOREM The time has come to consider some basic questions concerning Fourier coefficients of Lp functions; for instance, we are interested in identifying among the sequences { c j } with limljl+mcj = 0 those which correspond to Fourier coefficients of Lp functions and to make a more precise statement about their summability properties. The two results we have encountered so far are Icj(fil
s
IIfIIl
(1.1)
and
(1.1) states that the operationf + { c j ( f i }is of type (1, a), i.e., it maps L ( T ) into I " ( 2 ) ; (1.2) states that it is of type (2,2). Since the interpolation results in Chapter IV do not cover this case, we first attempt to restate the problem at hand as one to which they do apply. Following Zygmund we introduce the weighted sequence spaces l ; ( Z ) = 1; and wk-lr(Z) = wk-1;. 1; consists of those sequences { c j } so that the quantity IIcjll,: = (xjlcjlpp(j ) ) ' l P < 00,0< p < co (with the obvious interpretation when p = a),and wk-1; consists of those sequences { c j } for which there is a constant c > 0, so that, if OA = { jE 2:lcjl > A}, then AP p ( j ) s c, all A. An important example is p p (j) = (1 + 1j l ) - p ;note that I" G l:, whenever p > 1. With this notation (1.1) and (1.2) can be restated as follows: if to each f E L + L2,f 1 cjeiix, we assign the sequence {cjp2(j)-''2}, then this mapping is bounded from L2 to lt2and from L into wk-l,,.
xjEoA
-
142
1. Paley’s Theorem
143
Only the weak-type statement requires proof; this is easy since 0,= { j E 2 : lcjlpz(j)-l’z> A} G { j E 2: ~ ~ ( j 0.
+
+
If A( s)/ s increases and we set
then A. is convex, increasing, 0 at 0, and nontrivial (in fact A, is positive and finite in the same set as A); A. was introduced by Jodeit as the regularization of A. Moreover, A,(s) s A(s) s A0(2s). Functions which have the properties of A. are called Young's functions; cf. 5.2 in Chapter I. It is often convenient to use A rather than A, because, for example, min(A(s), B ( s ) ) satisfies (i)-(iv) but is not in general convex when A and B are convex. Let A(s)/s increase. The Young's complement of A is given by
a
A(u)= SUP(US- A(s))
(1.10)
sao
and the inequality su
s A(s) + A(u)
(1.11)
is called Young's inequality. The inverse of A is defined on [0,0O] by A-'(u) = inf{s: A(s) > u } ( i n f 0 = +a). (1.12) A-' is positive and finite for u > 0, A-'(oo) = 00, A-' is nondecreasing and right-continuous, and A-'( u ) / u decreases. Moreover, A(A-'(u)) s u s A-'(A(u)), u 3 o (1.13) and A(s) = sup{u: A-'(u) < s}, sup 0 = 0. If p and q are conjugate exponents, then the relation prototype of s s A-'(s)A-'(s) d 2s.
S ' / ~ S ' / ~=
s is the
(1.14)
For our purpose there is a special operation to mention, namely,
mR(s) = sup(l/A(l/u)),
s = 0, s > 0,
(1.15)
s 0.
1,
ek'f(X)I dx < 00 for
2. FRACTIONAL INTEGRATION
-
We have seen that iff E L, f C qei'", co = 0, then the Fourier series of its indefinite integral is Co l j + o ( l / (ij))cjevx.It is therefore natural to consider also the "fractional" analogue of the above formula, namely,
+
1 1-cje'ix,
o < < 1. (r
Ijl"
j+o
We see at once that (2.1) is the Fourier series of a function with the same integrability properties as f, since by Proposition 4.4 in Chapter I11 there is a summable function g with cj(g) = ljl-", j # 0. But we expect a better behavior from the series in (2.1), and, to get an idea of what this might be, we investigate whether we actually know something else about g. Consider the integrable function h ( x ) = IXI-~, 0 < 77 < 1. We note that its Fourier coefficients are given by the even sequence
'I
Cj(h)= -
?r
[O,r)
-cosjxdx xl)
and since the integral in (2.2) has a limit L # 0 as Ijl+ 00, we see that essentially cj(h) L./lj\l-q for ljl large. Thus g ( x ) lxlU-' near the origin, and in this case g E L'( T ) for r < 1/(1 - a).Therefore by Young's convolution theorem (cf. 4.30 below), (2.1) is the Fourier series of an Lqfunction, with l / q > l/p - a,whenever f E Lp(T) and 1 d p < l/a; this still is not sharp. The precise statement is the following: for 0 < a < 1 consider the Riesz fractional integral I, given by
-
-
I J - ( X ) = I,-dt,Itl1-a
x E T.
(2.3)
151
2. Fractional Integration We then have
Theorem 2.1 (Hardy-Littlewood, Sobolev). Assume that f E Lp(T), 1 < p < 1/a. Then Iuf E Lq(T ) , l/q = l/p - a and there is a constant c = c( a,p , q ) independent off such that
IIIafllq
(2.4)
CllfIlP.
Proof. Since IZuf(x)l d Z u ( l ~ ) ( xwe ) , may suppose t h a t f a 0. For 0 < 77 < T write
say. To estimate J note that + ( t ) = ~ f ~ a - l , y - l , lis~ (integrable, f) and, if + , ( t ) = ( l / T ) + ( t / T ) , then J = 77a j , f ( x - t ) + , ( f ) dt. Thus, by (a simple variant of) Proposition 2.3 in Chapter IV, J
C177uMf(X),
(2.6)
where c1 = 2/a is independent off: To bound K , note that since (1 - a ) p ' > 1, Holder's inequality implies that
where c2 = (2q/p')'/". Therefore, adding (2.6) and (2.7), we see that
Iuf(x)
C(rl"Mf(X)+ 77'-1~pllfllp~.
(2.8)
Now we seek to choose 7 to minimize the right-hand side of (2.8), a good choice being that 77 which makes both summands equal, i.e., ' 77 = ( I l f l l p / ~ f ( x ) ) " .
(2.9)
Note that this choice of 77 is all right as long as it does not exceed T ; however, if it does, it is because M f ( x ) c ~ - ' / ~ J ( f l l ~and for these x's, by setting q = T in (2.8), we see that I.uf(x) c cllfll,.
(2.10)
On the other hand, (2.9) gives that Iaf( x ) c c I1fll
X)
(2.11)
VI. Paley's Theorem and Fractional Integration
152
Since we prefer not to have to determine which estimate applies, we just add (2.10) and (2.11) and obtain that for all x in T I A X )
s ~ c l l f I l ~ " M f ~ x > '+ - "llfll,).
(2.12)
This is all we need to complete the proof. Indeed, since (1 - a p ) q = p , by (2.2) it readily follows that
IIIafIIq
~(IIfII~pIl"Mfll~-" + Ilfllp),
and since by the maximal theorem IIMfll, complete.
S
(2.13)
cllfll, for 1 < p , the proof is
Theorem 2.1 is best possible in the sense that an estimate of the form IIZ,Jllr s cllfllp implies r d q. This is readily seen as follows: for f(t) = ~ ( - , , ~ ) and ( t ) 1x1 d q/2 we note that
whence
and this can only hold as q + O+ provided that ( l / r ) + (Y B l/p. We must still consider p = 1 and p = l/a. The estimate (2.11) suffices to handle the case p = 1 quite easily, namely, Theorem 2.2. Supposef E L( T) and l/q = 1 - a ;then Imf E wk-L4(T) and there is a constant c, independent off; so that for A > 0
I{Lf>All Furthermore, iff B so that
E
(2.14)
C(llflll/A)q.
L In L( T), then IJ E Lq(T) and there are constants A,
IIZfll,
A+
j
lf(t)lln+lf(t)l dt.
(2.15)
T
Prmf. To prove (2.14) we may assume that A > 2~llf11~, where c is the constant in (2.12) above, for otherwise the conclusion is obvious. Then {IJ > A} E { ~ l l f l l : M f ' -> ~ A} and by the maximal theorem
I{Lf> A l l =S cl{M > ( ~ / ~ k f ~ ~ ? ) ~ ' ( ~ - ~ ) } ~ d
cllfll
llfll
Y/('-a)A
= c( llfll I l A ) q *
2. Fractional Integration
153
Also by (2.12), Theorem 5.3 in Chapter IV, and Young's inequality uOul-O's au (1 - a)u (cf. 5.2 in Chapter I), we get IIZOfllq s C ~ ~ f ~ +~ CllfIll ~ ~s cllfll~ ~ ~ +f cllMflll ~ ~ and ~ -we mare done.
+
Whereas the result concerning the weak-type is sharp, namely, there is an integrable function f so that I O f &Lq, l / q = 1 - a ( f ( t )= l / t ln(l/t)l+'/q for 0 < t < f = 0 otherwise will do) the L In L + Lq result is not, cf. 4.28. We consider next whether there is a maximal operator which in some sense controls the Riesz potentials as the Hardy-Littlewood maximal operator controls the Hilbert transform. First note that for f 3 0 and ?T>6>0
therefore a natural candidate for the task at hand is (2.16)
with 71. = a.The continuity properties of M , are readily established. Indeed, assume that 1 d p s 1/77 and j,lf( t ) p dt = 1. By Holder's inequality we see that
s
IZIn-lip(
[ If( I
t)lPdt)'/q
for any 0 < E < 1 since the integral s l . By choosing l / p - 7, we get at once from (2.17) that
E
=
1 - vp, or
(2.17) E/P
=
M,f(x) s M(Jfp)(x)'/"-"
(2.18)
I{M,,f > h}l
(2.19)
and consequently d
~/h'/('/~-"),
i.e., M, is a sublinear operation of weak-type ( p , q ) with l / q = l / p - 77, 1 d p d 1/77. This result can be improved to give the type ( p , q ) estimate for 1 < p s 1/ v ( p = 1/77 we already have) and this requires, of course, an interpolation result. Theorem 2.3. Let 0 < po s qo s co,0 < p1 s q1 s m, po < p l , qo # ql. If the sublinear operator T defined in Lpo(X)+ L p l ( X )is simultaneously of weaktype (PO, 40) with norm s c o and of type ( p l , q l ) with norm s cl, then T is
154
VI. Paley's Theorem and Fractional Integration
Of type (p7 9)s where l / p = ( l - V ) / p O + 7/p17 l / q = - 7 ) / 9 0 + 7/91 with norm S C C ~ - ~0C 1 and f is integrable and positive, then
the best constant c is ( p / ( r - l))p.State and prove a similar result for sequences. 4.5 If q < 2, then F2 f[o,tl s-” sin s ds decreases in (0, T ) and the same s-”(l - cos s) ds if q < 3. (Hint: Elementary calis true for t”-3 I[,,,, culus.) 4.6 If A, 3 0, p > 1, s > 0, and c < sp - 1, then (4.1)
implies n=l
Moreover, if c > -1, then (4.2) implies (4.1). (Hint: Sum by parts and invoke Hardy’s inequality for sequences at the appropriate places.) 4.7 If the An’s are the Fourier sine coefficients of the continuous function g, 1 < p < c o , l / p < < ( l / p ) + l and A , > O , then I x - a I - ” x ( g ( x ) - g ( a ) ) E LP[O,r) for every a, 0 s a < P, if and only if nP“- p -2 (C“,,k h k ) ’ < 00 (or equivalently 1 nPT-2(CsP_nh k ) P < 00). (Hinr: We do the necessity first; assume x - ” g ( x ) E Lp[O,P]. By 4.1 and 4.3 above, C n”-lhn converges. Now if E > 0,
E
s-”
sin ns ds
I 1I =
n”-’
(sn,xn)
s-“ sin s ds
I
=
O(n”-’),
161
4. Notes; Further Results and Problems
I(,,,,
I(,,,, s-“
uniformly in E and x. Hence s - “ g ( s ) ds = 1 A,, verges uniformly in E, and by letting E + O+ we obtain
sin ns ds con-
”
and each term on the right-hand side of (4.3) is positive. Applying 4.4 to the left-hand side of (4.3) we get that
1 (1 I A,x-’
C03971
s-”
sin ns d s ) pdx < a,
[O,X)
which implies that also
j[,,,(~
nAnx-’(nx)A-2
I
s-” sin u du
[O,nx)
Y
< a.
(4.4)
Now, we only decrease the integral in (4.4) by replacing the sum by Cfl=”, and then replacing ( nx)s-2 s-” sin u du by its minimum (here use 4.5); ~ ~ ~ , d x < co. Conversely, for any N > 0, thus, ~ [ o , f f ) ( nA,x’-”)P
I[o,nx)
and both terms are easily estimated. There is a corresponding statement for cosines.) 4.8 Assume that A,, decreases to 0 and denote by f either the Fourier cosine or sine series with coefficients A,, (cosine and sine results involving Lp integrability are usually equivalent because the series are conjugate to each other). Then for 1 < p < 00 and - l / p ’ < r] < l / p , x - ” g ( x ) E Lp[O,P ] if and only if 1 np”+p.-2AI:converges. (Hint: We deduce the cosine version from the sine version of 4.7 above; the sine version is obtained similarly. To show the necessity, assume for simplicity that A. = 0, then F(x)=
I
f ( t ) dt =
1C I A , sin nx
C0,X)
and by4.4, x - ” - ’ F ( x ) E Lp[O,P ) . Thus by4.7 1 nPT-2(1F=,hk)’ < 00, which gives the desired conclusion since the hk’S decrease. To prove the sufficiency notethat 2 f ( x ) sin x = C(A. - A,+z) sin((n 1 ) x ) ;therefore, wemay invoke 4.7 again (with 17 + 1 in place of 71 there) and obtain x - ” - ’ ( x f ( x ) ) E Lp[O,P ) , i.e., x - “ f ( x ) E Lp[O,P ) , provided that
+
VI. Paley’s Theorem and Fractional Integration
162
converges. But since C;=,lAk - Ak+21 = A, + A,+l we are done.) Assume f(x) is a positive, decreasing function in [0, T ) , 1 < p < 00 and -l/p’ < r] < l/p. Furthermore, assume that the An’s are either the Fourier cosine or sine coefficients off: Show that n-qplAnlPconverges if and only if xpq+p-2f(x)pis integrable over [0, T ) . (Hint: Dualize 4.8.) 4.10 Assume w(x) > 0 for x E T is such that for all A > 0, J{W(X)>*) w(x)-’ dx c A/A2, with A independent of A. If 1 < p 6 2 and { c j } E lP, then there is f e L(T) such that c j ( f ) = cj and 4.9
(j I f ( ~ ) I ~ w ( xdx)) ~ - ~ c C(II c ~ ~ ) ’ / ~ . 1/2
T
Also if 2 c q < 00 and f
- C cjeVx,then
(XI cj14)
d
(jTlf(x) pw (x)”-’ dx) l/q.
Can you think of “rearranged” statements such as Paley’s theorem? The Hausdorff-Young theorem and Paley’s theorem are not equivalent; indeed, let co = c1 = cP1= 0, cj = (ljl 111Ijl)-~/~, ljl S 2. Then C c;” = 00, but C c;”l j14/3-2< 00. for each 4.12 There is a continuous function f so that C lcj(f)12--s= E > 0. (Hint: Putf(x) =, :,I jp22-jI2(4(x)- &,(x)), where the 4’s are the Rudin-Shapiro polynomials introduced in 5.38 of Chapter V. The result, which, in particular, shows that there is no Hausdorff-Young theorem for p > 2, is Carleman’s, the example is from Katznelson’s book.) 4.13 The Hausdoe-Young inequality is best possible in the sense that if ‘ 00, then r p’. (Hint: Take a look for f E Lp(T ) , 1 < p < 2, ~ l c , , ( f ) l< at the function ]XI-”, 0 < r) < 1, which is in L P ( T )provided that r]p < 1, and its Fourier coefficients cj l j [ - ( l - q ) ,which are in I‘ only when r(1 77) 1.) 4.14 Assume that f E L In L( T), f C cjeiix,and show that there are constants A, B such that 4.11
-
’
-
c&1/1 1+
S
A+B
I,
(f(x)(ln+(f(x)(dx.
Is the conclusion still true if we replace lcjl by c . in the above sum?
+
4.15 I f f - 1 cjeVxand lcjl < l/ljl 1, then e A t f l lis l integrable, provided 0 < A is sufficiently small. (Hint: Use Hausdoe-Young to estimate llfll,, p = 2,3, . . . .) 4.16 Suppose { A j } E MPq, 1 s p c 2, 1 < q d 2. Show that I(1+
~j~)-q‘~p‘-E~Aj~q’ < 00 for each E > 0. (Hint: There is an Lp function with c j ( f ) ljl-’/p‘-’, n # 0. Since we assume that 1 Ajcj(f)eVxE Lq(T ) we may invoke the Hausdorff-Young theorem.)
-
163
4. Notes; Further Results and Problems
Assume that T is a sublinear operator simultaneously of weak-type (2,2) and of type (1, a), in both cases with norm c 1. Let B( t ) = b(s) ds, where b is positive and nondecreasing and A(t) = ( c o , t ) a ( s ) ds, where a is continuous and strictly increasing from 0 to 00 with s. Show that if t f[o,t) b(s)(ds/s2) d 1/2d(2/t), for t > 0, then B(ITf(y)l/2) dv(y) 1 whenever A(If(x)l) dp(x) c 1. (Hint: d is defined in Theorem 1.6; the result is from Jodeit-Torchinsky [ 19711.) 4.18 Assume that A(s) is as in Theorem 1.6, that {nk}is a fixed Hadamard sequence, i.e., nk+Jnk 3 A > 1, and that I,A(lf(x)l) dx d 1. Show that limk+OD snk(Ax) = A x ) a.e. (Hint: Cj MR(lCj(snkW - gnk(jI)l/2)< 00 (cf. 7.25 in Chapter IV).) 4.19 The real Laplace transform F of f given for x > 0 by F(x) = e-mf(u) du is a linear mapping simultaneously of types (1, 00) and (2.2). 4.20 The "local" control that M,f exhibits over Z,f in (2.20) may also be expressed in terms of the so-called "good-A inequalities," about which we will have more to say later on; these results are quite important in the understanding of the weighted norm inequality problem. In this setting, the good-A inequality assumes the following form: Let Z c T, A > 0, and, for positive constants E, S, let E = {x E I: Z,f(x) > EA, M,f(x) C SA}. Then there are constants E,, and k, depending only on a,so that iff is a nonnegative function and Z contains a point where Z,f(x) C A, then IEId Put g = fX21, h = f - g. We may k(S/E)l/('-a)lZl for all E > E ~ (Hint: . assume there is a point t E Z so that Maf( t ) d SA, for otherwise E is empty and there is nothing to prove. Also observe that by Theorem 2.2 there is a constant c (depending only on a) so that
4.17
Ico,t)
I,
I,
I[o,m)
Next let s E Z be the point where Iaf(x) < A. Since for some constant L > 1 and for x E Z and y iZ 21 we have that 1s - yl C Llx - yl, it is immediate to see that Z,h(x) s L'-"Z,f(s) C LA. If we pick now so = 2L, then for E a so the set {x E Z: Z,h(x) > ~ h / 2 is } empty. This result is from MuckenhouptWheeden [ 19741.) 4.21 Assume that f 2 0. Then for 0 < a, S < 1, IIZasfllr d c ~ ~ f ~ ~ ~ - s ~ ~ Z a f ~ where l / r = (1 - S ) / p S / q , 1 < p < q < 00. (Hint: As in (2.5) bound the integral defining Z,sf(x) by J + K ;estimate K by v)7e(6-1) Jlt-xpTf( t)lx - tl"-' dt d va(s-l)I a f(x) and minimize. This result and the next two are from Hedberg's paper [ 19721.)
+
:,
VI. Paley's Theorem and Fractional Integration
164
Suppose that f s 0, then IIIaS(ff)llrs c ~ ~ f [ ~ ~ - ' ~ ~where Z a f ~0~ < ~ ,a, 6 < 1 , O 0,
Once we show that
is dominated by the right-hand side of (4.5) we are done, and this is not hard to do. The result is O'Neil's [1966]; however, in the particular case of the fractional integral, it had already been noted by Zygmund.) 4.29 Let 0 s l / q i = pi =S l/pi = ai < 00, i = 0, 1, a. f a l , Po # P I , and let y = E X + y be the equation of the line passing through the points (ai, pi). Assume that the generalized Young's functions A, B are given by A ( u ) = b ( s ) ds, with a and b monotone and further a(s) ds, B ( u ) = assume that, if M = max(qo, 4 , ) and rn = min(qo, q l ) , then B ( u ) / u M decreases and B ( u ) / u " increases and
I
CO?)
( B ( s ) / s " + ' )ds = O ( B ( u ) / u " )
VI. Paley's Theorem and Fractional Integration
166
and
( B ( s ) / s M + 'ds ) = 0(B(u)/uM). Then if B-'( u) = A-'( u E ) u Ythere , is a constant c, independent off, so that J, B(ITf(y)l/c)d v ( y ) s 1 whenever 1 , A ( f[ ( x ) l )d p ( x ) s 1. (Hint: Set the monotone function z - l ( s ) = B-'(A(s)'/').) 4.30 (Young's Convolution Theorem. Lp * L' E Lq, l/q + 1 = l/p + l/r. (Hint: Fix f E Lp(T ) , 1 < p < 00, then convolution with f gives a bounded mapping from L into Lp and from Lp' into L"; now interpolate. Other proofs give a better constant.) 4.31 More generally, wk-LP * L' E Lq, same range of p, q, r's as in 4.30. 4.32 Assume that l/p + l/q > 1, 1 < p, q S 2; then Lp * Lq is not included in U,,,L", l / r + 1 = l / p + l / q . (Hint: For k > 0 put a,, = l/(k + 1)2/p2(k+')/p' for 2k 6 n < 2k+', and extend a, as an even sequence for all n's and similarly for {b,,} with q in place of p. Since 1uPnlnlp-2, 1bzln1q-2 < a there are functions f C u,eim E Lp and g 1b,,eimE L4. However, since ~ ( a n b , , ) S l n ~=S00- Zfor each s > r, f * g cannot be in L". This example is from Quek and Yap's paper [1983].) 4.33 If a > O , O < E < 1, 1 < p < &/a,1 s q < wand f E L" and M E I p E f Lq,then IJ E L' and 11 I, f 11 g cII M E I p11f"I" Ilfll;-"/". (This result is from Adams' work [ 19751.) 4.34 Show that M i = 9Lp, 1 < p G a, i.e. { A j } E Mi if and only if there is a function C#J E Lp(7') so that cj(C#J)= Aj?.allj . (Hint: The sufficiencyis obvious; as for the necessity, if A C Ajevx,then IIA * K,,l[ps C ~ ~ for K , , ~ ~ ~ all n. What does the statement imply concerning A:?) 4.35 The space of multiplier sequences from LP(T) into C(T) is g ~ " , l / p l/p' = 1, 1 s p < 00, and so is M%.What does the statement say about A%?
-
-
+
-
CHAPTER
VII Harmonic and Subharmonic Functions
1. ABEL SUMMABILITY, NONTANGENTIAL CONVERGENCE We refer here to yet another classic, and very important, summability method. This method requires the identification of T with eD, the boundary of the unit disk D = { z E C :Iz( < 1) in the complex plane. Given 0 s a < 7 ~ / 2we define the set Cn,(O) with vertex at 1, that is, e”, and opening a as the convex hull of the disk of radius sin a and { 1}\{ l},
u W O )
This set corresponds to the notion of “nontangential” approach to 1. It is readily seen that points z in Cn,(O) satisfy the condition 1 (1- zJ/(1- 121) S 2 max(l/(l - sin a),l/cos a).Thus it is equivalent, and often simpler, to consider instead the “cone”
r,(o) = { z E D : 11
- zl/(i
-
l z l ) c a},
a a 1.
(1.1)
We are now ready for
Definition 1.1. Given a
2
1 and a function f(z) defined in D, we say that 167
VII. Harmonic and Subharmonic Functions
168 \
f converges (Abel) nontangentially of order lim
Z+
i,zer,(o)
(Y
to L as z + 1 provided that
f(z) = L,
and we denote this by lim,,,f(z) = L ( A , ) . When a = 1 we call the approach “radial,” for then z takes values on the radius joning 0 to 1. Similarly, in the case of series we have Definition 1.2. Let { c i } be a sequence of complex numbers. We say that the m series cj is (Abel) nontangentially convergent of order a ( 2 1) to s, and denote this by CJys0cj = s(A,), provided that for f(z) = cjzJ, limze1f(z) = s(A,). The usual algebraic properties hold in this case as well. For instance, if so and s1 are finite numbers a n d x cj = so(A,),C dj = sl(A,), then also C(cj + d j ) = so + sl(A,).
CJzo
It would be reassuring to know that convergent series also converge nontangentially to the same limit; in fact more is true. Proposition 1.3. Suppose
CJco cj = s(C, 1).
Then also
,:C,
cj = s(A,),
lS(Y 0 so that If(z) - SI S E provided z E r,(O) and 11 - zI s 8. Indeed, combining (1.4) and (1.5), it readily follows that
c m
f(z) - s
+
= (1 - z)' (jro
(j
+ l ) ( q - S ) z i = I + J,
j=N+l)
say. Now, since m
m
j=N+1
j=O
1( j + I)+ s C ( j +I)$ =-( I -1 r)"
if we choose N so that laj - SI s & / 2 a 2for j that for z in ra(0),
3
N
+ 1, then it readily follows
Now that N has been fixed, we note that
j=O
provided S is small enough. We are interested in applying Proposition 1.3 to Fourier series, more specifically to Lebesgue's theorems 4.2 in Chapter I1 and 6.3 in Chapter 111. Since the expression appearing in the Definition 1.2 is one sided, given f E L, f C cjeij*,we introduce the notations
-
C o ( t )= co,
G(t)= c-je-l* + cje'",
j s l
(1.6)
j 2 1.
(1.7)
and eo(t) = 0,
c(t) =
(-i)(-cjeKij'
+ cjeij*),
Proposition 1.3 asserts that actually
j=O
and m
ej(t) = Hf(t)(A,)
almost every t in
T.
( 1 -9)
j=O
We may, and do, assume that (1.8) and (1.9) hold simultaneously. More precisely, if z = rek E ra(0),then for almost every t in T (1.10)
170
VII. Harmonic and Subharmonic Functions
and (1.11)
We would like to unravel (1.10) and (1.11) and express them in terms of the original cj's. First, since r,(O) is symmeric, that is, z E r,(O) if and only if Z E ro(O), and as is readily seen
then for almost every t in T the following is true: (1.12)
Similarly, by (1.11) now, and for the same t's,
"+" statement
Whence, by subtracting the "-"statement in (1.13) from the in (1.12) and rearranging the expressions involved, we obtain
= f(t )
- iHf( t ) ,
a.e. in
T.
(1.14)
Similarly, by adding the "-"statement in (1.12) and the "+" statement in (1.13) and rearranging the expressions involved, we obtain
=f(t)
+ iHf(t),
a.e. in
T.
(1.15)
We have thus arrived at one of the most interesting and important results in this chapter, namely, Theorem 1.4. Suppose f E L( T ) , f
- C cjeGr.Then for almost every t in T (1.16)
and (1.17)
2. The Poisson and Conjugate Poisson Kernels
171
Prmf. (1.16) follows by adding (1.14), (1.15), and (1.17), by subtracting (1.14) from (1.15), and invoking the symmetry of r,(O) to change -x intox. H
2. THE POISSON AND CONJUGATE POISSON KERNELS Expressions (1.16) and (1.17) in Theorem 1.4 correspond to the convolution off with the functions P ( z) and Q( z) with (absolutely convergent) Fourier series given by m
z = re",
O < r < 1 (2.1)
and W
(-i)(sgn j)cjrlj'evL,
P(z) = Q(Z) =
z
=
reiL,
o 0 and all 0 < r < R < R , , and constants c, a,x independent of r, R Then the same conclusion holds with the expression on the right-hand side replaced by
and a constant c which depends only on p.
CHAPTER
VIII Oscillation of Functions
1. MEAN OSCILLATION OF FUNCTIONS We introduce in this section a maximal function which has become extremely important in various areas of analysis including harmonic analysis, PDEs, and function theory. The spaces generated by this maximal function are also of interest since, in the scale of Lebesgue spaces, they may be considered an appropriate substitute for L“(T) and beyond. Of course the notion of “appropriate” is a matter of personal choice, but from our point of view an appropriate substitute for La(T) is a space which is preserved by a wide class of important operators such as the HardyLittlewood maximal function and the Hilbert transform and which can be used as an end point in interpolating Lp spaces. In this sense the JohnNirenberg class BMO( T ) we consider below fits the bill. We introduce this space as follows: for f E L( T) let
where c above varies over the complex constants and I is an interval containing x, (I) s 27r. This definition can actually be simplified. Suppose that f is real valued. Then so will be the constant c which minimizes the integral in (1.1). By elementary considerations we expect c to be among those values for which
Since the integrand above equals 1 for f ( t) > c and -1 for f ( t) < c we have, in particular, that I{f > c}l = I { f < c}l. This actually means that any 199
VIZZ. Oscillation of Functions
200
such constant c verifies simultaneously I{f > c}l d 111/2 and I{f < c}l s \Z)/2. In other words c = mf(l)is a median value of f over 1. These considerations can be formalized,and extended to the case whenfis complex valued by introducing the median values mr(I) = r n R e f ( I )+ imImr(I)..Is there, however, a simpler way to choose c? The answer is contained in the next result. Proposition 1.1. If constant c
h
denotes the average of f over I, then for any
Proof. Since
If(t) -hl
If(t) - CI + Ic -hl =z If(t) - CI + (l/lZl) M Y ) - CI dY,
the conclusion follows at once upon integrating over I. Corollary 1.2. For f E L( T) and Z
c T we have
If(t) -&I
dt s 2inf c
I,
If(t) - CI dt.
(1.3)
In view of this corollary we may redefine the “sharp” maximal function in (1.1) by the equivalent expression (1.4)
where I is an open interval containing x, 111 c 2 ~ Clearly, . M # f is a measurable, subadditive function. Let now
Ilfll*
=
IIM”fllm
(1.5)
and put BMO( T) = BMO = {f E L( T ) : Ilfll* < 00). This is the John-Nirenberg space of functions with bounded mean oscillation. Endowed with the norm given in (1.5), BMO becomes a Banach space provided we identify functions which differ a.e. by a constant; clearly, llfl1* = 0 for f(t) = c a.e. in T. Bounded functions f are in BMO and Ilfll, =z 2\lfllm; however, observe that ))xI)I* = 4. On the other hand, does BMO contain unbounded functions? The standard example that this is the case is f(t) = lnltl, It1 < T ; we sketch the proof of this fact. Let I = (a,b) c T. We show that for an appropriate choice of c I ,
1. Mean Oscillation of Functions
20 1
-
which in turn implies that lllnl ](I* s 2. To prove (1.6) we consider three cases, namely, (i) 0 < a < b, (ii) -b < a < b, and (iii) the rest. In case (i), we pick Cr = In b and note that jI/lnltl - In bl dt =
I
(In b - In t ) dt
(46)
= (b - a ) - a(ln b -In a). 2
Therefore,
I I I jrllnltl -In
bl dt
=
(In b - In a ) b-a ’
1-a
and (1.6) follows at once since 0 < a < b. In case (ii) we may restrict ourselves to - b < a < 0 < b. Again pick cI = In b and note that ~rllnltl- In bl dt
=
I
llnltl - In bl dt
(a,-a)
+J
(In b - In t ) dt
=J
+ K,
(-a,b)
+
say. The above computation shows that K = ( b a ) + a(ln b - ln(-a)). As for J, since the integrand is an even function, it equals
2 lim E+O+
Thus J
I
(In b -In 1 ) dt
= 2(-a
In b
+ a In(-a)
- a).
(&,-a)
+ K = ( b - a ) + a(ln b - In(-a)) jIlln/tl - In bl dt
=
1 - (-a)
and (In b - In(-a)) (b + a ) b+a (b-a)‘
(1.7)
Since a < 0, the right-hand side in (1.7) is s l , and (1.6) holds in this case as well. The remaining cases can be reduced to either (i) or (ii) since we are dealing with an even function. Now that we know that BMO functions are not necessarily bounded, the question is how large thay can be; we take another look at lnltl. Fix (0, b) = Z c T and consider those t ’ s in Z where lnltl is large, i.e., consider
oA= { t E I: llnltl - c I / > A},
A
> 0,
Cr = (lnl.I)I. We are interested in OA for large values of A. Clearly, OA = {t E I : t > eA+‘1}u { t E I : t < e-”+‘I}. Obviously, the first set in 0, is
where
empty for A large, and for those A’s we get
10Als I{t E I : t < e-”+‘I}I
= epAeCI.
Vlll. Oscillation of Functions
202
I,
By Jensen's inequality, ecr s ( l / l l l ) eln'dt = I4/2 and consequently ISAI c f\I1 e P A . The remarkable fact is that a similar estimate holds for arbitrary f ' s in BMO and I c_ T. More precisely, we have Theorem 1.3 (John-Nirenberg Inequality). Assume that f E BMO and I G T. Then there are constants c , , c2 > 0, independent off and I, so that lit E 1:I f ( t )
-hl >
< cle-cz'/llfll* 1 1 1
(1.8)
for all A > 0.
Proof. By replacing f by (f-h)/llfIl*if necessary, we may assume that = 1; we must then prove that
h = 0 and Ilfll*
lO,l
= I{t E I : I f ( t ) l > A}[
s cle-czAIII.
(1.9)
This is achieved by the use of the Calder6n-Zygmund decomposition. First, since ( l / l l l ) Irlf(t)l dt s Ilfll* = 1, we may invoke the Calder6n-Zygmund decomposition for f( t ) X r ( t ) at level 2. We thus obtain (a first generation of) open, disjoint subintervals { I ; } of I such that (i) I f ( t)l s 2 a.e. in I\U I,!, (ii) 2 < ( l / I I J ) J , ! l f ( t ) ldt s 4, and (iii) I;II s f l , I f ( t ) I dt = f I U ( l / I 1 I ) I r I f ( t ) I d t ) c f l ~ l .
c
Next we consider each I,! individually. To simplify notations, fix such an interval, call it I', and consider the function (f(t ) - fil)xIl(t ) . Since
we may invoke the Calder6n-Zygmund decomposition of (f(t ) - hi(t)xll(t ) at level 2 and obtain (a second generation of) open, disjoint subintervals { I ; } of I' such that (i) I f ( t ) s 2 a.e. in I'\U 13, dt c 4, and (ii) 2 < (l/ll;l) t)(iii) ~11j'ls f~ J , ; ( f ( t )-hi1 dt s fl1'1.
I,?lf(
Moreover, considering all 1 " s now we also have (i') for each I ' , If(t)l 6 I f ( t ) -hi1 + Ifrll (ii') CalljJ1;I s f Call 11JI'I (4)2111-
2.4 a.e. in I'\u I;, and
We continue with this process and obtain at the nth step a family of open, disjoint subintervals { I ; } of In-' such that
If(
t)l
c 4n
a.e. in
I"-'\U 17
(1.10)
203
1. Mean Oscillation of Functions
(1.11) These estimates are all we need. Suppose first that A > 4 and let n 2 1 be the integer so that 4n < A S 4(n 1). By (1.10) it readily follows that 0; E {t E Z:I f(t)l > 4n} E U Zjn and by (1.11) that
+
l'Al
(f)"lrl*
(1.12)
Moreover, since 2-" = e-" ln and A s 8n, we note that 2-" s c2 = (In 2)/8. Thus by (1.12)
eCC2"
loA]s e-czAIZJ
with
(1.13)
in this case. If, on the other hand, 0 < A s 4, then
loA[s IZ)s cle-c2hlzl
(1.14)
provided that cle-c2A2 1 when A S 4; to ensure that this occurs we set 1 -- e4(ln2)/8 = & > 1. By combining (1.13) and (1.14), we have that in all cases I{t E I: If(t)l > A}[ S & e-(1"2/8)A111, as we wanted to show. The converse to the John-Nirenberg inequality also holds, namely,
Proposition 1.4. Assume thatf E L( T) and that there are constants c l , c2 > 0 so that I{t E I: If(t) -hl> A}I S c1e-'2^lI( for Z c T and A > 0. Then for 0 < c < 4, eclf(t)-frl E L( T) and (1.15)
Proof. We note that the left-hand side in (1.15) equals c
I{t E I : If(t) [0,m)
-fll>
A}\ecAdA s ccllZl
J
e-(c2-c)A
dh,
[OF)
and we are done. Corollary 1.5. Suppose that f E L( T). Then f E BMO with norm and only if
Ilfll*
if
Proof. To obtain the necessity of the condition observe that by the JohnNirenberg inequality and Proposition 1.4 we have (1.17)
VIII. Oscillation of Functions
204
where cl, c2 are the constants in Theorem 1.3 and 0 < c < c,. From (1.17) it follows at once that for p 3 1
JI
and we have finished. Conversely, if (1.16) holds for anyp 3 1, it also holds for p = 1 and the desired conclusion follows from the John-Nirenberg inequality. 2. THE MAXIMAL OPERATOR AND BMO
In this section we prove that the Hardy-Littlewood maximal operator is bounded in BMO.
Theorem 2.1 (Bennett-DeVore-Sharpley). Assume f E BMO. Then Mf E BMO and there is a constant c independent off such that IIMfll* cllfll*. (2.1) Proof. Since M f ( x ) = M ( l f l ) ( x ) and 11 Ifl S 2llfll, (cf. 6.12), we may assume that f 3 0. We must then show there is a constant c, independent of the subinterval J E T and f; so that
Fix an interval J and for each x in J divide those intervals I c T containing x into two families according to their relative size with J; more precisely, let $ , ( x ) = {Ic T : x E I and I c 35 n T} and B;*(x) = {Ic T : x E I and I n (T\3J) # 0).If we set now
and
it clearly follows that M f ( x ) = max(Fl(x), F2(x)). Furthermore, since j , ( M f ( x ) - (Mf),) dx = 0, if 6 = { x E J : M f ( x ) > (Mf),}, we readily see that
205
2. The Maximal Operator and BMO
Thus, if we set %, = {x E 6: F , ( x ) s F 2 ( x ) }and 4!L2 = 6\%,,we may rewrite the right-hand side of (2.5) as 2 2 (F~(x - )(MA,) dx = A + B, (F~(x - )(Mf),) d X +-
m I,,
I JI
I,,
say, and (2.2) will hold for J provided we show that that for some absolute constant c, A, B 6 cllfll*. (2.7) We consider A first. Let now I denote the interval 3 3 n T Since h s infxszM f ( x ) s inf,,, M f ( x ) , then clearly fi s (Mf),, and consequently we may invoke the Calder6n-Zygmund decomposition of fxz at level (Mf),. We thus obtain a sequence of open, disjoint subintervals ( 4 ) of I verifying
(i) f(t ) s (Mf), a.e. in Z\U 4, (ii) (w3,< (1/141) Iz,f(t)dt c 2(Mf),, all j , (iii) U 4 E I, and there is an additional property we emphasize: If 1; is the “ancestor” interval corresponding to 4, i.e., I; is the dyadic subinterval of Z which when subdivided gave rise to 4, then (iv) IZJ = 2141 and (l/lI;l) j z ; f ( t )dt s (Mf),. We may then consider (a variant of) the Calder6n-Zygmund decomposition g + b of fxz obtained by putting and
= C f r ; x r , ( t )+f(t)xr\ur,(t) i
b(t) =
C(f(t)-h;)xz,(t). j
By (i) and (iv) above it follows that g(t)s
(MA,
in contrast to the usual decomposition where g ( t ) s 2(Mf),. Corollary 1.15 and (iii) and (iv)
(2.8) Also by
llfll:.
(2.9) This is all we need. Indeed, by (2.3) we readily see that F , ( x ) s M ( f x z ) ( x )s M g ( x ) M b ( x ) and by (2.8) and (2.9) that ClJl
+
I,,
F l ( x ) dx d
I,,
M g ( x ) dx
+
M~(x dx)
(2.10)
206
VIII. Oscillatiqn of Functions
Thus passing the first summand from the right to the left-hand side in (2.10), we see at once that A S cllfll*, which is the A statement in (2.7). To bound B we actually prove the stronger estimate F2(x) - (Mf3.r S
~llfll*,
x
(2.11)
E q 2 .
Fix x in q2and let Q be any interval in $72(x);clearly IQI 2 IJI. Consider now the subinterval Q u J of T, 1Q u JI S 21Ql. As above we note that fQu J s ( M A and consequently
Taking sup over Q E 9 2 ( x ) we obtain (2.11), which in turn implies the B statement in (2.7), and we have finished.
3. THE CONJUGATE OF BOUNDED AND BMO FUNCTIONS
As we remarked in 2.7 of Chapter V, if f ( x ) = X ~ ~ , ~ ~ then ( X ) ,f This statement holds in general. In other words,
E
BMO.
Theorem 3.1. Supposef E L"( T). Then? E BMO( T) and there is a constant c independent off so that IIjII, S cllfllrn. The proof of this theorem is essentially contained in that of Theorem 3.3, which we do shortly. However, this is an excellent opportunity for the reader to prove this result directly. At any rate, before we consider our next result we need an observation concerning BMO functions. Proposition 3.2. Suppose f E BMO and I E T, then lf2r
-&I
6
2llfll*,
-&I
5
2kIIfII*,
21
T
(3.1)
and M2kr
2 k ~ T.
Proof. Since (3.2) follows from (3.1) on account of the observation k
- f i ~ CISzjr -hj-lrL
~ 2 . r
j=1
(3.2)
207
3. The Conjugate of Bounded and BMO Functions it suffices to prove (3.1). But this is easy since
zs
2
[ 2 r l f ( t )-f211 dt s 2llfll*.
We are now ready to prove Theorem 3.3 (Spanne, Stein). Assume that f E BMO. Then f there is a constant c, independent of f; such that
E
BMO and
Ilfll, s cllfll,.
(3.3) Proof. Given an interval I c T, we show that there is a constant c, = c( I, f ) so that
;IrIf(t)
- CII dt = C l l f l l * ,
(3.4)
where c is independent of I and f;this clearly implies llfII* s 2cJJfJJ,and we have finished. First, since f E BMO, then f E L2 and f is a well defined, L2 function. Fix an interval I and put
f ( t ) = ( f ( t )- h ) ~ 2 r ( t + ) ( f ( t ) -h)~7-\2i(f) +h
= f d t ) + f 2 ( t ) +h, say. Since (h)" = 0, we have f ( t ) = f l ( t ) + f 2 ( t ) , and it suffices to show that (3.4) holds with f replaced by fi and fi,with suitable constants c,,, and q 2 ,respectively. The estimate for fl, with cr,, = 0, is immediate since
say. By Corollary 1.5, A s cllfll*, and, by Proposition 2.2, B
with c independent of I andf; as we wanted to show.
2llfll,. Thus
208
VIII. Oscillation of Functions
To estimate F2(f) let
I,
denote the center of I and put
where the integral is absolutely convergent, since f2 vanishes in a neighborand the notation k( t) = - l / ( 7r2 tan( t/2)), hood of I. With this choice for c I , ~ we have
” ”
The integrand of the innermost integral in (3.6) was denoted k(t, x, t,) in (1.10) of Chapter V and estimated in (1.13) there by c(lIl/lx - t,12). Thus / I and x the right-hand the innermost integral in (3.6) is of order c ~ I ~-~tIl2 side of (3.6) does not exceed a multiple of
I
~ 1 x -1~
1112
I / I X-
trl’) dx.
(3.7)
T\2I
If now denotes the largest integer k so that 2kI G T, the integral in (3.7) is bounded by
k=l
say. We examine each Ak more closely; since
4. Wk-Lp(T) and Kj-. Interpolation
209
Whence replacing the estimate (3.9) in (3.8), by (3.7) we see at once that (3.6) is dominated by
In other words
(3.10) (3.4) follows by adding (3.5) and (3.10), and we have finished. Theorem 3.4. Assume f E BMO.Then there is a constant c, independent of
J; such that IIH*fll* c cllfll*. Proof. By Theorem 2.13 in Chapter V, H * f ( x ) c(Mf(x) + M(Hf)(x)), and the conclusion follows at once from this on account of Theorems 3.3 and 2.1.
4. Wk-LP(T) AND K,. INTERPOLATION
Let Z c T and (f -h) E Lp(Z), 1 s p < 00. If ( 4 ) is a finite collection of open, disjoint subintervals of Z, then by Corollary 1.2 and Jensen’s inequality it readily follows that
The consideration of a converse to (4.1), which also corresponds to taking l p rather than lm norms on the sequences {(l/lrjl)jI,lf(t) -hll d t ) , as was done in the definition of BMO,leads to Theorem 4.1 (John-Nirenberg).
Suppose thatf
E
L ( I ) and that for some
l I f l l ; in our case, since
p 2 l/lZl’’p will do. For such p, invoke the Calderh-Zygmund decomposition at level p and obtain (a first generation of) open, disjoint subintervals Zj of Z such that
s 2p
+ 2p = 22p,
we may invoke the Calder6n-Zygmund decomposition of this function at level 2’p. We then get (a second generation of) open, disjoint subintervals {Z;} of Z’ such that (i) If(t) -fill s 2’p a.e. in z’\U I,? dt s 23p, all j (ii) 2*p < (l/lZ;l) j I j f ( t ) ) II;lf(t) -hi1 dt =s (1/2’p) (iii) ClIj’l ( 1 D 2 p .C
jAf(t)
-hi1 dt.
We would like to rewrite (i) and (iii) in terms off taking into consideration all the 1 ” s . It is readily seen that we have (if) If(t)l s I f ( t ) -fill
+
=S 22p
+ 2p s 23p, a.e. in z’\U I,?,
and summing over all j ’ s corresponding to all Z”s also (iii‘)
Callj’slz;l
s (1/22p) C a l l I l . s I I ~ l f ( f )-511 dt.
We take a closer look at the sum on the right-hand side of (iii’). With l/p + l/p’ = 1, we note that it does not exceed
211
4. Wk-Lp(T ) and K,. Interpolation
Thus (iii’) actually states
c IIi’l
all j ’ s
c (1/22P)( ( U P ) j,lf(t)ld t ) l / p ’ .
Clearly, this process may be iterated as we did inthe proof of the JohnNirenberg inequality: in general, and in the kth step we invoke the CalderhZygmund decomposition for the function (f- & k - i ) x , k - i at level 22(k-1)p and we get a collection of open, disjoint subintervals { I ! } of Zk-’ so that
If(
t)l s
22k-1 p
zk-’\UZ!
a.e.in
(4.5)
and
where l / p
+ l/p’ = 1 and
4(p,k) = ( 2 2 ( k - 1 ) p ) ( 2 2 ( k - 2 ) p ) l / ~ ’ ( 2 2 ( k - 3 ) p ) ( 1 / ~ ‘ ) * . . . p ( l / P ‘ ) k - ’ To complete the proof all we need is a good estimate for the right-hand side of (4.6); note that 1 22/~’(1:=li/(P’)’-’) --
4(p,k) - (22(k-1)p)I + l / ~ ’ + ~ ~ ~ + ( l / p ’ ) “ ~
Moreover, since, :,I
(4.7)
j/(p’Y-’ = 1/(1 - l / ~ ‘ =) pz, ~
c MP’Y
c m
00
=P
and
MP’Y = P/(p’)k,
j= k
j=O
the right-hand side of (4.7) can also be written as 2Zpz/p’(22(k-1) p)P/(P‘)’/ (2Zk-1 ) p)P.
(4.8)
Furthermore, since ( k - l ) / ( ~ ‘c) ~cp = c for all k 2 1, the numerator in (4.9) does not exceed ~ p ~ / (and ~ ‘consequently )‘ the right-hand side of (4.6) is bounded by
We pick now p = 2/1Z)1/pand observe that with this choice,
(fI,!
pplp’
-
r)l dt 6 2P/P’K f -= c,
which finally obtains that for this choice of p the right-hand side of (4.6) is bounded by c/(22(k-l)p)P.
(4.10)
212
VIII. Oscillation of Functions
We are now ready to prove (4.3). First assume A > 2p and let k be the largest integer 31 so that 2*&-’p < A. In this case, and on account of (4.5) we have that 0,c U I f , where the union is taken over all j ’ s and k’s. By (4.10) then (4.11)
10Ald ~ / ( 2 ( ’ ~ - ’ )sp c)/~A P ,
which is what we wanted to show. It only remains to consider the case A s 2 p = 4/lI11/p;but now AplOAls APIIl d 4 and we are done. Not only is Theorem 4.1 of interest in itself but it also has important applications. The one we consider here is to interpolate with BMO as an endpoint space. Theorem 4.2 (Stampacchia). Suppose T is a linear operator which is bounded in Lpo(l), 1 d po < 00 and which is of type (00, *) there, i.e., ; T maps L p ( I )into itself, po < p < 03. (1 T ~ ( ( B M o ( ~I()( f ( ( ~ - ( ) , )then Proof. Let (4) be a finite partition of disjoint subintervals of I, and put
We claim that T is a sublinear mapping simultaneously of types ( p o , po) and (a),@) with norm, in each case, bounded independently of the particular partition of I. This is easy to check in the co case since II~fll, s (1 Tfll*s cllflloo (all norms are taken over I) and for po note that since the 4 ’ s are disjoint, by (4.1) we have
c 2( 1, l T f ( t ) - (Tfirlpodt)l’poccllTfllpo+ ~
z
~
1
’
~
Moreover, since IIll/polTflrs (ITfllpb, also I.fl, s cllfllpo,and our claim is verified. By the Marcinkiewicz interpolation theorem in Chapter IV we obtain that T is bounded on Lp(I),po < p < co,with norm bounded independently of the partition ( 4 ) as this is the case for p = p o , 00. Moreover, since the 4 ’ s are disjoint we readily see that
and consequently for each f in L p ( I ) ,K , s cllfllp, with c independent of JByTheorem4.1, Tf- ( T f r isinwk-LP(I)withnormscllfllpforpo
l =s I f b ) -hl + Ih -f(v)l
A + B, (5.1) say. We only consider A in (5.1), since the estimate for B is identical. We construct a sequence of subinterval {Jk} of J which tends to x as follows: put J1= J, J2 the left half of J1, i.e., the half which contains x, lJ21 = ilJ1l, and so on. Clearly, for k 2 2, If(x)
c I&+, -h"I
=
k-1
A c If(x) --hJ+
= A1 + A2,
(5.2)
n=l
say. Next note that
c 2c2
IJI" c 2("-1)oL O0
= C,C,lJI".
n=l
Thus, substituting this estimate in (5.2) we see that A c If(x) -hJ+ c,c21JIp, all k > 2. (5.3) Now by Theorem 2.2 in Chaper IV, for almost every x in Z we have limn+mlf(x)-hkl= 0 and consequently by (5.3) we readily see that A cc21x - yl" for almost every x in Z. A similar estimate holds for B, provided, of course, that y is also a point where Theorem 2.2 of Chapter IV applies. Therefore by (5.1) we get that If(x) - f ( y ) I
cc21x - yl",
for almost every x, y
in
I.
(5.4)
Since it is evident that f may be redefined so that (5.4) holds everywhere in Z, the proof is complete. An interesting application of the above result is
Theorem 5.2 (Privalov). Assume 0 < a < 1 and f E Lip,( T) = Lip,. Then f E Lip, and there is a constant c, independent of so that
IlfllL
cllfll*,.
(5.5)
215
5. Lipschitz and Morrey Spaces
Proof. The proof is similar to that of Theorem 3.3. In the notation of that theorem, and by (iv) with p = 2 above, we note that
and since I f 2 1 - f l l s 2 ~ ~ 2 ~ 1 1we 1 "get that
Similarly, since
Ak in (3.8) of that theorem is readily estimated as follows: first,
1
I f ( x ) - L k + l I I dx s ~4(2~+'lIl)'+" 2k+lr
and 2k111 IfZk+lr -hl s CC4k2k111'+",
we see at once that Ak S
JJf2Ct)
c4
-
+
C C , ( ~ ~ ( ~ - ' )k2-k)111"-1.Therefore,
- Cr,2I dt
also,
CC41111+ol,
and we have finished.
~ ~ f ~ ~ A a ,
There is yet another important family of spaces defined in terms of oscillations and we will consider it here briefly since, in some sense, it provides a glimpse of what, in addition to Lip,, lies beyond BMO. Given 1 s p < 00 and 0 d 7 < 1 + p let Zp,,,(T ) = ZP,? denote the class of f~ L ( T ) such that j r l f ( t ) - h I p d t s cPIIIv,
all
I E T.
(5.6)
The smallest constant c in (5.6)above is denoted by and is called the Endowed with this norm, SP,? becomes a Banach space norm off in ZP,?. provided we identify functions which differ by a constant. For each fixed p these spaces coincide with familiar ones for some values of 7 ;for instance, if 7 = 0, Zp,,?Lp; if 71 = 1, ZP,?BMO, and if 1 < 7 < 1 + p , then ZP,,, Lip(q-l),p. In the range 0 < 7 < 1 they are called Morrey spaces and it is possible to show that also j r l f ( t ) l p d ts cl1l" for each I c T. Some relations among these spaces, such as ZP,? E Zp,,,,if p 3 p , and (7- l ) / p = ( 71- l ) p l are quite easy to prove. How about interpolation properties and the action of the conjugate operator on these spaces? The reader may think about these questions, or consult Peetre's work [ 19691.
-
-
-
216
VZZZ. Oscillation of Functions 6. NOTES; FURTHER RESULTS AND PROBLEMS
In the early 1960s John and Nirenberg introduced a space, which they called BMO, in order to be able to use the fact that a function which can be approximated in every subcube Z of a cube Zo in the L mean by a constant a, with an error independent of Z differs also in the Lp norm from a, in Z by an error of the same order of magnitude, 1 < p < a;this is of course one of the meanings of the John-Nirenberg inequality. The proof we give here of this result follows ideas of Calder6n (see Neri [1971] and Garnett [1981]). An interesting description of the role of BMO in the theory of elasticity, as well as the introduction of the local sharp maximal function M$ considered in problem 6.1, is in John’s work [1964]. Further Results and Problems 6.1 Given a measurable function f defined in T and x in T we put
M&f(x) = Sup, inf, inf{A 2 0: I{y E Z: If(y) - C J > A}I < sIII}, where Z is an interval containing x and 0 < s s f (this restriction on s is necessary since for s > f, M & f = 0 for any function which assumes only two values); some properties of this function, such as M{,(f+g)(x) s 2(M&f(x) + Mc,/,g(x)) are easy to show. An important relation is the following: if x E Z, then I{y E Z: If(y) - mf(I)l > 4M&f(x)}l < sill; prove it. (Hint: To show the last relation, given E > 0 let c, = a + ib be such that l{y E Z: If(y) - c,l s M&f(x) + E } J 2 (1 - s)lZl; then
KY
E
1:IRef(y) - 4 > MCSf(x)+ &}I < s l I l
and a similar inequality holds for IZ,f(y) - bl as well. Thus a ( M & f ( x ) + E ) S Re(mf(I)) s a + ( M & f ( x ) + E ) and similarly for b and Im(mf(Z)), and Ic, - m,(f)l s Mc,f(x) + E. Let now I, = { y E Z: If(y) - c,I 2 Imr(Z) - c,!}, Z, = Z\Z,; since it is readily seen that l{y E 1,: If(y) - mf(Z)l > 2(M$f(x) + &)}I = 0, the choice E = M&f(x) completes the proof. Clearly an equivalent statement is l{y E I: If(y) mf(Z)l > 4 inf, M&f)l < sill; this result and the next 5 are from Stromberg’s paper [ 1979al.) 6.2 Let f be measurable, 0 < s s f, Z E J, IJI s 2 k l Z ( . Then Imf(Z) - mf(J)I s ck inf, M&f: (Hint: Cf. (3.2).) 6.3 Assumefis measurable Z E T, 0 < s < f,and A, p > 0; then if I mf(Z)I A there is a collection ( 4 ) of disjoint, dyadic subintervals of I such that no interval 4 is totally contained in {y E Z: M c , f ( y ) > P } , A
p } ) . This statement corresponds to the Calder6n-Zygmund decomposition off and is proved similarly. 6.4 Suppose f is measurable, 0 < s 6 and I E T. Then for P, A > 0, I{y E I : I f ( y ) - mf(I)I > PII s cle-CB/*lIl+ I{Y E I: ~ C , f ( y>) AIL where c, c1 are absolute constants. (Hint: This statement corresponds to the John-Nirenberg inequality and is proved in a similar fashion. An interesting variant of this result is the following: for J; /3, A as above, s sufficiently small, and 0 < p < a,I{y E I: If(y) - mf(I)l> P,
M C S f ( Y )6 APII
w-=/* Ilf - mf(I)ll”L(r,/PP,
6.4 corresponds to p = 0. This result is due to Jawerth and Torchinsky [ 19851.) 6.5 Suppose f is measurable, 0 < s s f, and M & f E La( T). Then f E BMO and Ilfll* ~ ~ M & f One ~ ~ mof. the interesting features of this result is that our original assumption only involved the measurability, and not the integrability, of J 6.6 Suppose f~ L, 0 < s < 1. Then M&Hf(x) G (c/s)M#f(x), x E T. (Hint: Fix I and let x E I; then put fl(y) = (f(y) - mf(I))xzr(y) and f 2 ( ~ = ) ( f ( y ) - ~ ~ ( I ) ) x T \ ~ I ( YSince )Hf(x) = Hfi(x) + Hf2(x), we treat each term separately: it is easy to show that for y in I IHf,(y) - 6 c,M#f(x) and by the weak-type (1,l) property of H that Al{y E I: IHfl(y)> l A}[ s cM”f(x)lIl.) 6.7 A subadditive operator T is of weak-type (1,l) and maps L“ into BMO if and only if for every interval I, f E L + L“ and A > 0, I{y E I: I T ~ ( Y ) - mTf(I)l> 1 ,11 s c l e - c A ~ lmin(lI1, l f ~ ~ - ~ ~ f ~ ~ l / A(Hint: ). The proof of the necessity makes use of the variant of the John-Nirenberg inequality given in problem 6.4. This result is from Jawerth-Torchinsky [19851, where also another necessary and sufficient condition is given, namely, I{M;,Tf> A}l 6 cll{Mf> ch}l, cl, c constants which depend on T but not on f and h > 0.) 6.8 Prove that BMO is complete. (Hint: It suffices to show that if N j T f n ( t )dt = 0, n 3 l,andCllf,II, < a,thenlim,,,C,,=,f, existsin BMO; it is not hard to show that the limit in L actually coincides with that in BMO.) 6.9 Show that Ilflll + Ilfll* gives an equialent norm in BMO, where now functions are identified when they coincide a.e. in 6.10 Does there exist a BMO function f so that f E L”( T)? (Hint: If f E L2(T), then = -f c o ( f ) . ) 6.11 Show that llnlxllP& BMO for p > 1; also X ( ~ , ~ , lnlxl ( X ) E! BMO. 6.12 Show that M”(lfl)(x) G 2M#f(x). (Hint: By Proposition 1.1, jrlIf(t)I - Iflrl dt G 2IIIMt)I - IhIl dt 6 2I,If(t) - f i I d t ) . 6.13 Suppose f E BMO and show that If(t)l* also belongs to BMO for 0 < LY 6 1; this is in contrast to Problem 6.4.
-
CI
5
7
+
VZZI. Oscillation of Functions
218
Suppose f is real valued and show that for each real constant c, M#(max(f; c))(x) s ;M#f(x) and M#(min(f; c))(x) s ; M # f ( x ) . (Hint: max(f( t ) , c ) = i(f(t ) c ) + t ) - CI and min(f( t), c ) = t)
6.14
+
- ilfW - Cl.1 6.15 Suppose f E BMO( I), 11f11* CI
;If(
$If(
+
I,
= 1, f(t ) dt = 0; then f can be written as g + h where h ( t ) = C,:, ajxIJ(t),the 4 ’ s are dyadic subintervals of Z, (ail S c, all j , and llgllms c,; c, c, are absolute constants. (Hint: We take a closer look at the John-Nirenberg inequality; with the notation of Theorem 1.3, let fl =fxI\“I,, fi = C ajxJJ,aj = f J J , (aj[s 2. Next apply the CalderbnZygmund decomposition to eachf’ = (f- uj)xJJand obtainf:,fi as before; repeat this process and set g equal to the sum of all the A’s, h equal to the sum of all the A’s. This result is related to some of Varoupoulos [1977].) 6.16 Here is another way to build up BMO functions: given a sequence ( 4 ) of dyadic (not necessarily disjoint) open subintervals of T, we say that a sequence of functions ( a j }is “adapted” to the 4 ’ s provided (i) aj vanishes off 34, (ii) l a j ( x ) - aj(t)l s Ix - ti, all j , (iii) laj(x)l 6 cl, IuJx)l s c2/lIl a.e., all j . If, in addition, Cj. Jll,ls c31ZI, for all Z G T,then for any adapted sequence of functions we ’have IIC ajll* S c(cI + cz)c3. (Hint: Given Z, separate those 4’s so that 31, n Z # 4 into two families, 8, = (1,: 141S 111) and 8, the rest, and put
C
f(t) =
aj(t),
g(t) =
C
aj(t)-
i;JJ €32
J ;1, E JI
Clearly,
To estimate A = (1/1Z1) j , l g ( t )
- c,l
dt, let
and note that each Ek contains at most A
d
f
k=O
j:JjEEk
1’
I
~
1
cubes ~ 3 and bound
l u j ( t ) - aj(xj)ldt,
where xi
E
4.
I
xi
What this means is that we chose C, = aj(xj).The result is due to Garnett and Jones [1982] and it admits a converse.) 6.17 Suppose f E BMO. Then for 1 d p < 00
with c independent of f: Conversely, i f f € L and (6.1) holds for any 1 s p < 00, thenf E BMO. (Hint: The proof of the first statement is similar
219
6. Notes; Further Results and Problems
in spirit to Theorem 3.3. On the other hand, suppose (6.1) holds for p = 1, say; if z = re”, I = interval centered at t with 111 = 27r(l - r) and x E I, then Pr(x- t) 5 c/lIl,and,consequently, (l/lZl) j I l f ( x )-f* P,(t)ldx s c, all Z. Note that if (6.1) holds for p’s < 1, we may also infer that f E BMO by invoking 6.5 above.) 6.18 With the same notation and assumptions as in 7.14 of Chapter I11 and 7.29 of Chapter IV, show that s* E BMO and IIs*ll* s c(C Irk(’)’/’. (Hint: Given I G T, let N be an integer such that 2 7 r / h ~ + < ] 111 s 27r/A~. There are two cases. First, if such an N does not exist, it is because (11> 27r/Ao, and the bound for (l/lIl) I,s * ( t ) dt obtains at once from 7.29 in Chapter IV. If, on the other hand, such an N exists let t be the center of Z and put cz = sup{ls,(t)l,. . . ,IsN(t)l};we claim that in this case
To show (6.2) we make repeated use of the following fact: if u,, u, 3 0 and l u n - unI s 1 7 then IS UP,,^^ u, - supnaou,l s 1. Let then a ( x ) = sup{lsO(t)l, * * * IsN(t)l, ( S N ( f ) + RN+I(X)I, * . * I S N ( t ) + RN+dX)19* . -1 where R,(x) = C;=,rkfk(Ag); by 7.29 in Chapter IV I s k ( x ) - S k ( t ) l s c(~Irkl2)’/’if k s N, x E I, and, by the above remark, I a ( x ) - s*(x)l S c(C(rk(’)’/’ as well. If now p ( x ) = SUP{IRN+I(X)I,. . . ,IRN+k(x)I,.. .}, then I a ( x ) - cI\ s p ( x ) and finally Is*(x) - czI s p ( x ) c(Clrkl’)’/2 where ((1/1Z1) p ’ ( x ) dx)’/’ s c(CIrkl2)*/’;thus (6.2) holds and we are done. The result is Meyer’s.) 6.19 I f f € BMO, then there exists a constant E > 0 such that 9
+
I,
sup (I{x IET
E
Z: I f ( x ) -hl>
A}~/ I ‘ZI)
s e-*/‘,
(6.3)
whenever A > h ( ~ , f )indeed, ; by the John-Nirenberg inequality, we have E = cllfll*, h ( ~ , f= ) clllfll,. Let now ~ ( f =) inf{s > 0: (6.3) holds} and show that
Moreover, also by the John-Nirenberg inequality there is 7 > 0 such that
< 00,
(6.4)
and let ~ ( f =) sup(7 > 0: (6.4) holds}. Show that ~ ( f =) l / ~ ( f ) .These results are from Garnett and Jones’ work.
220
VZZZ. Oscillation of Functions
6.20
Let 4 be a positive, nondecreasing function defined on [ 0 , 2 ~and ] put 1)=
IZl4(lZl)
j p x )
and finally BMO, = {f E L: l[fll*, tially due to Spanne [ 19651.
-hl
dx,
llfll*,,
= SUP 4(J;1) I= T
< 00). The following results are essen-
if Z E J and
If Z(x, r ) = interval centered at x, length 2r, then Ilf -&., CP(A r ) ; If f~ BMO, and TJ(x) =f(x + s), s 3 0, then
,)I(* s
[If- ~ f l l *
=
O(p(s));
If 4( t ) / t is nonincreasing, then f(x) = j(lxl,l,4( t ) / t dt E BMO, (this is in a way the largest function in the space). + ( t ) = t“, 0 < 77 < 1, and suppose that M$f(x) = supxpr4(J;Z) E Lp(T ) , 1 < p < 1/77; then MZZf E L‘( T ) , l / r = l/p - 77 and there is a constant c independent off such that IIM”fll,. s cI(M$fll,. 6.22 Let Z = [0,1]. We say that a function g is a “pointwise multiplier” of BMO,( Z) provided that gf E BMO,( Z) whenever f E BMO,( Z) and there is a constant c independent off such that Ilgfll*., S cllfll*,,. Assume ) / sthen g is that 4 ( r ) / r is nonincreasing and put $ ( r ) = 4 ( r ) / ~ ( , . , l ~ ( sds; a pointwise multiplier of BMO,(Z) if and only if g E BMO,(Z) n L“(Z). (Hint: Since p(J I ) s ~(r)~~f~~*.,, by 6.20 it follows that for any Z with 1Z1 = r < we have 6.21 Assume
1
-f,g,l
r ~)f(x)g(x)go
dx c 4(r)llgll, s
llfll*,9 + lhl
c 4 ( r )llfll*,,(
1
j l g ( x ) - gll dx
llgll* + Ilgll BMO,,).
Conversely, let Z = Z(t, I ) be an interval as above, let f be the function introduced in 6.20 (iv), and put h(x) = max(f(x - t ) , J ( ? . , , 4 ( s ) / s d sthen ); h is continuous and Ilh l*, G cllfll*,,. Since gh E BMO,, a simple computation shows that g is as it should be. For +(t) = 1, the result is due to Stegenga [1976] and for more general 4’s to Janson [1976].) 4 ( s ) / s 2 ds; one easily checks 6.23 Given 4 as in 6.20 let $ ( r ) = r that $ is nondecreasing, $(r ) + 0 as r + 0 , 4 ( 2 r ) < 2$( r ) and $(r ) 3 4 ( r ) . Show that, iff E BMO,, then Hf E BMO4 and there is a constant c independent off such that 11 Hfll *,J s ~ l l f l l * , ~ . (Hint: The result is Peetre’s [ 19661 and the proof is similar to that of Theorem 5.2.)
6. Notes; Further Results and Problems
22 1
Sarason introduced the class VMO(T) of functions of vanishing mean oscillation in T, consisting of those f s in B MO(T ) for which j , l f ( x ) -hl dx = 0. Obviously, VMO contains every conlim~r~+.,,(l/~I~) tinuous function in T, and since, as is easily verified, VMO is a closed subspace of BMO, it also contains the BMO-closure of C ( T); in fact, VMO is precisely that closure and in many ways VMO bears the same relation to BMO as C does to L". There are many reasons to consider VMO. Here is one: we are already aware that the conjugate f o f f € C ( T ) is not necessarily continuous (cf. 5.14 in Chapter V); however, it is in VMO. A quick proof of this result goes as follows. Iff E C ( T), then we can write it as the sum of a trigonometric polynomial and a function of small L" norm; by Theorem 3.1, f is then the sum of a trigonometric polynomial and a function with small BMO-norm. Thus is in the BMO closure of C ( T ) as we wished to show. For further applications the reader should consult Sarason [ 19791. 6.25 Prove that VMO(T) = BMO,( T ) . 6.26 f~ V M O ( T )if and only if lim,,,llf-f* Pr(-)ll*= 0. 6.27 With the notation of 6.18, show that s* E VMO. 6.28 Let f~ VMO. Then M # ~ C E( T ) . (Hint: It suffices to show that for A > 0, { M # f < A} is open; to do this, suppose M # f ( t ) < A and we will show that the same holds in an interval about t. Since f is in VMO, there l / ~-hl I ~ )dJyI) < M # f ( t ) ;now let exists S > 0 such that S u p ~ ~ ~ ~ ~ ( (If(y) J be an interval centered at t whose length is very small compared to 6, let x E J, and, finally, let I be an interval containing x. If 111 < 8, then (l/lIl) J I l f ( y )-hi dy S M # f ( x )< A by the choice of 8 ; suppose then that 111 2 6 and let K = I u J. Then K contains t and 6.24
u
Since IJI is very small compared to 111, the intervals K and I are almost the same and M # f ( x ) < A as well. The same proof works for
and this result has an important application, namely, f is an extreme point of the unit ball of VMO normed by 11 Mrfll,, 1 < p < a,if and only if MFf( t ) = 1. These results are from Axler and Shields [19821.) 6.29 BMO is the natural substitute for L" also in the case of fractional integrals. More precisely, if 0 < a < 1 and p = l/a, then I, maps Lp(T)
222
VIII. Oscillation of Functions
into BMO( T) and there is a constant c independent off such that 11 IJII* G cllfll,. (Hint: With the notation of (2.16), we prove that M ( I , f # ( x ) s cMJ(x). Fix I = I ( x , r ) and put f = fxZ1+ fxT\2r = g + h, say. Note that
1
=z c r F
thus I(I,f)rl c M J ( x ) and consequently ! r I L g ( y ) - (I,g)ll du crMJ(x). By the mean value theorem, we get that for t in I, II,h(x) I,h(t)( d cr (T\211y - xl"-'lf(y)l d y d cM,f(x), a.e. in t, and the right estimate for this term follows by integrating over I. In fact the ideas in this proof give a more complete picture of the general situation when f 3 0, for in that case we have that (i) I J E BMO if and only if M J E Lm and (ii) I,J M,f and M ( I J ) # have comparable norms in Lp, 1 < p < 00. These results are discussed in Adams [ 19751.) 6.30 Let f-1cjeiix, f~ Lip,(T), 0 < a < 1, show that cj = O(ljl-u IlfIl,,). This result is sharp, indeed, iff(t) = Cr=l 2-"" cos 29, then f~ Lip,(T), and cj - j - , , j = 2". (Hint: As in 5.16 in Chapter I we consider g ( t ) = CFi, d , , , ~ ( ~ ~ - ~ , , J twhere ), a, = 2?m/lkl, Ikl Z 0, n = 1 , . . . ,Ikl. Use that c k ( g )= 0 and the choice d,, =Aa"-,,a n ) to obtain the first assertion. Next, observe that if I = (a, b) c T, then m
1
f ( t ) -h = n = l 2-n,(cos2"' =
-
(sin 2"b - sin 2"a) 2"(b - a )
1 2-",A(n, t, I ) , n=l
say. Now IA(n,t, Z)I s 2, but we need a better estimate when 111 is small compared with 2". In that case, IA(n, r, I)I S c(2"lZl). This gives If(t) -&I clIl", t E I, and we have finished. Note that, if a = 1, the above estimate becomes If(t) -hl d clIl ln(2m/lIl) instead.) 6.31 LetfE Lip,, 0 < a < 1 and show that s.(X x) - f ( x ) = O ( n 7 In n ) as n + CD, uniformly in x ; this result cannot be improved. 6.32 Let f~ L. Then a,(J x ) - f ( x ) = O ( n - , ) if and only i f f € Lip,, O A}; 0, is open since to each x in 0, there corresponds an open cube I, containing x such that
and consequently I, c OA.In fact
0,=
u I,.
(1.3)
xeU*
We want to estimate p ( 0 , ) in terms of the p-measure of the (complicated) set on the right-hand side of (1.3); it is apparent that we need some control 223
ZX. Ap Weights
224
over this set. So, suppose, in addition, that p is regular; in other words, if % is p-measurable, then
If K is a compact subset of OA now, there are finitely many Zx’s,Zx,,. .. ,I ,, say, so that K s U ,; Ixj; in fact, we may avoid unnecessary overlaps by .,. However, even once this discarding any cube Ix, such that Ix, c Uj+kI is done we may still be left with quite a bit of overlap. To handle this we proceed as follows. Since we are dealing with a finite number of cubes, there is one with largest sidelength (if there is more than one just pick any); separate it and rename it Zl.Now, if any of the remaining cubes, say I, intersects I,,since sidelength Z s sidelength of Zl,it follows that I c 311, the cube concentric with Zlwith sidelength three times that of Zl;all these cubes Z can be discarded as well. We are thus left with a finite collection of open cubes, each one disjoint with Il .Repeat for this family the procedure used to select Zl,that is select a cube with largest sidelength, call it I,, and discard all other cubes which intersect it. After a finite number of steps we are left with a collection {II,.. .,zk} of disjoint open cubes so that K E 31,. Thus
u;=,
Under what circumstances can we replace p ( 3 4 ) by p ( 4 ) in (1.5)? This can be done for the so called doubling measures, namely those measures p for which p(21) s cp(Z), all open cubes Z, c independent of I. In case p is doubling then from (1.5) it follows that
where c is the doubling constant of p. But the 4 ’ s are special cubes; in particular they all satisfy (1.2). Whence combining (1.6) and (1.2), and since the 4 ’ s are pairwise disjoint, we get k
Finally, on account of (1.4), (1.7) gives
2. The Hardy- Littlewood Maximal Function and the mapping is of weak-type (1,l) with norm =sc have proved
225 2
. Summing up, we
Theorem 1.1. Let p be a nonnegative Bore1 measure in R" which in addition is finite on bounded sets, doubling, and regular. Then the mapping f + MJ is of weak-type (1, I), with norm O h P P ( { M - > A))
kPllfll%;.
(2.1)
What can we, then, say about p ? Simple examples show, for instance, that p cannot have atoms, but in fact much more is true.
IX. Ap Weights
226
Theorem 2.1. Assume p is a nonnegative Bore1 measure, finite on bounded sets and assume that for some 1 S p < co (2.1) holds. Then (i) p is absolutely continuous with respect to Lebesgue measure, i.e., there is a nonnegative, locally integrable function w such that d p ( x ) = w (x) dx. (ii) for all open cubes I and f E LP,(R”)
where c = cp,k is independent of $ (iii) ( A pcondition) w satisfies Muckenhoupt’s A p ( R ” )= Ap condition, or w E Ap;i.e., there is a constant c = cp,k independent of the open cube I so that
or
I
The infimum over the constants on the right-hand side of (2.3) and (2.4) is called the Ap, or A l , constant of w ; moreover, the Al constant of w is less than or equal to ck and the Ap constant of w is less than or equal to ckP-’, p > 1. A statement which depends on the Ap constant of a weight w rather than on the weight itself is called “independent in Ap.” Theorem 3.1 below is an example of this. (iv) (Strong Doubling) For each open cube I and measurable subset E of I, s c(lll/lEI)PP(E),
where c
=
(2.5)
cp,k is independent of E and I.
Proof. (i) Suppose the Lebesgue measure IE( of E is 0. We show that also p ( E ) = 0. By the regularity of the measures involved we may assume that E is compact and that, given E > 0, there is an open set 0 3 E such that p(0\E) < E. Let f(y) = X B \ ~ ( ~ ) ;then JE LP,(R”) and l l f l l ~ ; = p(O\E) < E. Next observe that M f ( x ) = 1 for x E E : indeed, to each x E E there corresponds an open cube I c 0, x E I, and consequently 1/111 j r f ( y ) dy = I(O\E) n 11/1I1 = 1 since IEl = 0. By (2.1) it then follows that p ( E ) S p ( ( M f 2 f})C 2PkPllfllPL:s CE,which gives the desired result as E is arbitrary.
227
2. l%e Hardy- Littlewood Maximal Function
(ii) Fix an open cube I and consider for f~ LL the quantity which we may assume >O. Since (1/111) jrlf(y)l dy =
If], must be finite for each 1,for otherwise (2.1) cannot hold unless p is the 0 measure. Thus, if we put 0 = { M ( f X r )> lflr/2}, by (2.1) and (2.6) it follows that p ( I ) S p ( 0 ) s k ” ( l / l f l ~ ) ” l l f x ~ lwhich l ~ ~ : , is equivalent to (2.2). (iii) and (iv) Since there is no a priori reason why w cannot vanish on a set of positive measure, we introduce the measure d v ( y ) = d p ( y ) + E dx, E > 0, to avoid unnecessary technical difficulties. Clearly, v is also absolutely continuous with respect to Lebesgue measure, d v ( y ) = u ( y ) dy, u > 0, and, more importantly, (2.1) also holds with p replaced by v with constant independent of E. Assume p > 1 first. In order to estimate u(y)-”(”-’) dy we note that it equals Ill/ullpL’, l/p + l/p’ = 1, which by the converse to Holder’s inequality may be estimated by
I,
Now by (ii), which also holds for v, it follows that for such f’s
and consequently
Unraveling (2.7) gives (2.3),that is, the Ap condition for u. Next we show that (2.5) holds for u; indeed, note that
which by (2.7) does not exceed c k v ( E ) ’ / P ( I I I / v ( I ) ’ / PIn) . other words
v(I) s ck”(IZI/IEI)”v(E).
(2.8)
IX. Ap Weights
228
Since the constant in (2.8) is independent of E, we may let E + 0 there and obtain that p ( I ) 6 ckP(III/IEI)Pp(E)as well; i.e., (iv) holds. Moreover, if there is a set E with IEl > 0 so that p ( E ) = 0, then by (2.5) p is the 0 measure since it vanishes on any open cube I containing E. Thus d p ( x ) = w ( x ) dx, w > 0 a.e., and the above argument may be repeated with w in place of u; this completes the proof when p > 1 . The case p = 1 is rather simple since by putting f = xE, IEl > 0, in (ii), we see that (IEl/lIl) d c k ( p ( E ) / p ( I ) )or equivalently
P(O/IIl C k ( P ( E ) / I E l ) , (2.9) which is (iv). Furthermore, by choosing E to be a sequence of open cubes converging to a Lebesgue point x of w so that w ( x ) ess inf, w, (2.9) gives (2.4), and the proof is complete.
-
3. A , WEIGHTS As we have seen, A , is a necessary condition for the Hardy-Littlewood maximal operator M to map L,(R") into wk-L,(R"), but is it also sufficient? Theorem 3.1 (Muckenhoupt). Suppose w E A , . Then M maps L,(R") into wk-L,(R"), with norm independent in A l . Proof. First note that if w E A , , then p is doubling with doubling constant s c ( A , constant of w ) ; indeed, since (1/1211) d p ( y ) C c ess inf, w s c ( l / l I l ) d p ( y ) , it readily follows that p ( 2 I ) c c p ( I ) , c s 2 " ( A , constant of w ) . Moreover, since
I,,
I,
we also have that M f ( x ) C c M J ( x ) , c = A , constant of w. Thus {Mf> A} E {MJ>A/c}, and by Theorem 1.2 Ap({Mf>A})s 3 ( A / C ) P ( { M J > Ale)) =s c IlfllL,. Some observations concerning Al are obvious: for instance, A , is the limiting Ap condition as p + 1+ and an equivalent way of stating Al is M w ( x )s cw(x)
a.e.
(3.1)
3. A, Weights
229
But what are the A, weights? Can we give some examples or even characterize them? As a first step we consider powers of 1x1, i.e., 1x1". When n = 1 and 71 > 0, by letting I = ( 0 , 6 ) we note that (1/6) j(O,b)X" dx = l6"/(7 + 1) + 00 as 6 + 00, whereas inf, x" = 0. Thus positive powers of 1x1 are ruled out, but how about negative powers? We must have -n < 7 C 0 for otherwise 1x1" is not locally integrable, but this is essentially the only restriction. Indeed, we have
Proposition 3.2. Suppose -n < 7 C 0. Then 1x1" E A,; more precisely, there is a constant c independent of I such that
Proof. Fix a cube I and let I, denote the translate of I centered at 0; we consider two mutually exclusive cases, to wit (i) 21, n Z # 0 and (ii) 2 1 0 n I = 0. In case (i) we have that 610 2 I and ( l / ~ I ~ ) ~ , ~ xC ~ " d x ( l / ( I ( )j6r01xI"dx c ~(ll"'", where cis a (dimensional) constant, independent of I; clearly, (3.2) holds in this case. Case (ii) is easier, for then 1x1 lyl for x, y in I; indeed, we have 1x1 c Ix - yl + lyl G clIll'" + IyI clyl, and the opposite inequality follows by exchanging x and y above. Thus IyI" G cinfr)xlg,all y E I, and averaging over y in I, (3.2) holds for case (ii) as well. H Next we consider functions which behave like (XI", -n < 7 S 0, and show that they also are A, weights.
-
Proposition 3.3 (Coifman-Rochberg). Let p be a nonnegative Bore1 measure so that Mp(x) is not identically 00. Then for each 0 s E < 1, Mp(x)" E A,, with A, constant which depends only on E. Proof. Recall that Mp(x) = S u p x E ~ ( l / ~ Z ~ ) For p ( I )a. fixed open cube Z we ,Mp(x)" dx by A" = (inf, M p ) " as follows: for each x in estimate (l/lI() I I we divide those open cubes Q containing x into two families by setting $I = {Q: IQI C I2Zl) and $* = {Q: IQI > 1211). Thus
= A(x)
+ B(x),
say. The estimate for B ( x ) is readily obtained; since for Q 3 Q 2 Z, it follows that
(3.3) E
$2
we have
and B ( x ) 6 CA
(3.4)
IX. Ap Weights
230
with c independent of p. As for A(x),let p , denote the restriction of p to 61, i.e., d p , ( y ) = xsr(y) d p ( y ) , and note that
A ( x )d w h ( x ) .
(3.5)
Thus on account of (3.3), (3.4), and (3.5) we get that
and it suffices to prove the desired estimate with M p replaced by M p l . But by (a simple variant of the Lebesgue measure version of) Theorem 1.1 and 7.5 in Chapter IV, we readily see that [I
1 MP~(X dx) S~ -c(wk-L norm of M p I ) E ( I I 1 - "
with c depending only on
I[I
E,
and we have finished.
The interesting fact is that the converse to Proposition 3.3 also holds, name1y,
Theorem 3.4 (Coifman-Rochberg). Assume w E A,. Then there are functions b and f and 0 d E < 1 so that (i) 0 < A d b ( x ) d B < 00 a.e. (ii) J E L,,,(R"), M ~ ( Xis )finite ~ a.e. and w ( x ) = b(x)Mf(x)'.
The proof of this theorem relies on the so-called "reverse Holder" pr (perty of w ; this property is of independent interest and plays an importaat role in the theory of weights.
Theorem 3.5 (Reverse Holder). Suppose w E A,. Then there is a positive number q so that
where c = c,, is independent in A , and independent of 1,but not, of course, of q ; we abbreviate (3.7) by w E RH,,,,.
3. A, Weights
23 1
Now, suppose that Theorem 3.5 has been proved. Then by (3.7) also E A, ,M ( w'+")(x) s cw(x)'+" a.e. and Theorem 3.4 holds with b(x) = w ( x ) / M (W ' + " ) ( X ) ' ' ( ' + " ) , f(x) = w(x)'+", and E = 1 / ( 1 + q ) . It thus only remains to prove Theorem 3.5, which we do forthwith. In order to assure that the various integral expressions we consider are finite, we introduce the function v ( x ) = min(w(x), N); clearly, v E A,, (A, constant of v ) s ( A , constant of w), independently of N: indeed, for a given I let A = inf, w and consider two cases, namely, (i) A 3 N and (ii) A < N. We then have wl+"
N s inf, v,
in case (i),
w ( y ) dy s c inf, w
s c inf, v, in case (ii),
thus proving the claim. We show (3.7) with w replaced by v first; let q > 0 and observe that
=A+B,
(3.8)
say. Clearly, B
(3.9) which is the right estimate. The bound for A is not so readily obtained, and in the course of the proof we must keep track of the various constants appearing to be sure that they only depend on the A, constant of v, and 77 of course. First observe that with the notation 0,= { y E I: v ( y ) > t } we have A = (1 + q )
j
S V,+"IIl,
t"lo;l dt
Cv1,m)
lSSlds)' dt
IX. Ap Weights
232
which is also of the right order. Next we show that for an appropriate choice of 77, D is dominated by the (finite) quantity (3.12)
which may then be passed to the left-hand side of (3.8) to obtain the desired conclusion. We consider the innermost integral in D first. It equals (3.13)
Now since t > 0 1 , we may invoke the (n-dimensional version of the) Calder6n-Zygmund decomposition of u at level t, thus obtaining a collection of open, disjoint subcubes (4) of I with the following properties (i) v ( y ) s t a.e. in I\U 4 and (ii) t s (1/141) u ( y ) dy < 2"t, all j.
I,
From (i) it is clear that { u > t} c_ does not exceed
u4 and therefore the integral in (3.13) (3.14)
Furthermore, since u E A,,
c inflJu, UIJ
c
since lj is a Calder6n-Zygmund cube.
{29,
Therefore by combining these bounds we get that ulJ=s(c infrJu)'-'(2"t)", all j , O < E < 1. Thus each summand in (3.14) is dominated by ct'lIjl(infIJ v)'-' s ct" jrJ u(y)'-" dy and (3.14) does not exceed (3.15)
We need one last observation: from the left-hand side inequality in (ii) it 4 G {Mu > t}, and, since u E A, and M v ( x ) 6 c v ( x ) a.e., follows that we also have U 4 G { u > t / c } . Consequently, (3.14) is bounded by
u
ct'
I
{V>t/C)
v(y)'-"dy,
0<E 1
233
and the same is true of (3.13). Whence
(3.16)
First fix 0 < E < 1 and then choose 77 > 0 sufficiently small so that cv(1 + q ) / (+~7 ) < this is clearly possible. Thus D is dominated by (3.12) and (3.7) holds with v in place of w there. This is a minor inconvenience since by Fatou's lemma
t;
'/I+"
6 lim inf N+m
( I,i
'/I+"
v(y)l+? d y )
and the proof is complete. H 4. A,, WEIGHTS, p > 1
As we have seen, Ap is necessary for the Hardy-Littlewood maximal operator M to map LP,(R") into wk-LP,(R") and a simple argument similar to Theorem 3.1 shows it is also sufficient. However, a stronger result holds. Theorem 4.1 (Muckenhoupt). Suppose w E A,, 1 < p < 00. Then M maps LP,(R") continuously into itself, with norm independent in Ap.
Proof. For a nonnegative function f in LP,(R") and an integer -00 < k < 00, put A, = { y E R": 2, < M f ( y ) S 2k+'} and let %k be a compact subset of A,; we estimate Mf(y)'d p ( y ) by cllfll;;:, where c is independent of f and the %k's, and depends only on the Ap constant of w. A simple limiting argument then gives the desired result. To each y E A, we assign an open cube Iy containing y so that
Iuq,
f ( x ) dx
( Oandnotethatp - E - 1 = ( p - 1)/(1+7 ) .Itisthen readily seen that
(4.14) and also
Whence, by multiplying (4.14)and (4.15),it follows that w,w:-" we wanted to show.
E
A,-,, as
5. Factorization of Ap Weights
231
(ii) Since w2 E A, from Holder's inequality (applied to (1/1Z1) w2(y)'/"'/~ ~ ( y ) d' y/) ~it' follows that for some constant c > 0 independent of I
I,
(4.16) Let w1 E RH,,,. Then by (4.16)
as we wished to show. (iii) Let w E RH,,,. Then by Holder's inequality
= c( IEl/lIl) ? / l + ? p ( I ) .
(iv) By (i) and the doubling property (iv) of Theorem 2.1, p(2kZ0) < C 2 n k ( P - - E ) cL(Zo), k 3 1.
(4.17)
Thus
and we have finished.
5. FACTORIZATION OF A,, WEIGHTS This section is devoted to proving a remarkable fact, namely, the converse to Proposition 4.3. Before we proceed with the proof of this factorization result we need some preliminary observations; we start with a definition.
IX. Ap Weights
238
Definition 5.1. We say that an operator T is admissible provided it verifies the following four properties. (i) (ii) (iii) A > 0. (iv)
There exists r, 1 < r < 00, so that T is bounded in L'(R"). T is positive, i.e., T f ( x )2 0 for every f in L'(R"). T is positively homogeneous, i.e., T ( A f ) ( x )= ATf(x) a.e. for each
+
T is subadditive, i.e., T(f+ g ) ( x ) S T f ( x ) T g ( x ) a.e.
Some examples of admissible operators include If(x)l, M f ( x ) , and, more important for our purposes, ( ~ ( ~ f l " / v " ) ( x ) u ( x ) "for ) ' ~appropriate " 1< p < m , o < 7 d 1. We verify this last example and in the process we find the necessary conditions on u for this to hold. First, we must find r, the choice r = p / 7 being a natural one; in this case we have IITfll: = J R m M(lfl"/u")(x)'/"u(x)dx and, if we assume that u E All", then this ) = cllfll: and expression may be estimated by c JRn ( l f ( x ) l " / " / u ( x ) ) u ( xdx (i) holds; the verification of (ii) and (iii) are immediate. Thus it only remains to check (iv). Fix x and let I denote an open cube containing x; then by Minkowski's inequality we readily see that
=G
M(lfl"/u")(x)""+ M ( l g y / u " ) ( x ) ' / q
and since I is arbitrary it follows that M ( l f + glp/u")(x)l'pd M ( l f l p / u " ) ( x ) l /+ p M ( l g l P / u " ) ( x ) l / pand , (iv) follows at once. An important property of admissible mappings T is that they also are cT-subadditive; more precisely, we have
Proposition 5.2. Suppose T is admissible and r is the index in pro ert (i) in the definition of T. If {A},f are L'(R") functions with limN+, =f in L', then T f ( x ) ?s CJ:, TJ(x) a.e.
R Y
hf. Since f = (f -
zJzl&) + xJT,&,from properties (ii) and (iv) it fol-
lows that N
N
j=l
239
5. Factorization of Ap Weights
Moreover, since 11 T(f-CJt,f;)llrG c l l f - C ~ , f i I l + . 0 as N + 00, there is a sequence Nk + 00 such that lirn,,,, T ( f - C z , f i ) ( x ) = 0 a.e. This is the sequence of N's we choose in (5.1), and by letting Nk + 00 there we get Tfi(x) a.e. that T f ( x ) S ZT=, We need one more definition. Definition 5.3. Given an admissible mapping T, we say that a nonnegative function w is in A , ( T ) if T w ( x )s c w ( x )
a.e.
(5.2)
We then have Proposition 5.4. Suppose that Tl and T2 are admissible mappings with the same r in (i). Then there exists a function 4 in L'(R") such that 4 is simultaneously in A,( T I )and A,( T2).
+
Proof. Put T = TI T2( T is also an admissible mapping) and let A 11 7'11, the norm of T i n L'(R"). For an arbitrary, nonnegative function g in L'(R") put 4 ( x ) = T j g ( x ) / A J We . show that this 4 will do. In the first place, 4 E L'(R") sinceCJz,((T'g((,/A's Tll/Ar)llgllr< 00. Moreover, by the a-additivity of TI we see that
(CJzo(II
This proves that 4 is in A,(T,) and a similar argument gives that 4 is in A1(T2)as well. W It is now a simple matter to prove the decomposition theorem for the Ap weights. Theorem 5.5 (Jones). Suppose w E Ap,1 < p < 00. Then there are weights w,,w 2 in A , so that w ( x ) = wl(x)w2(x)'-". Proof. Since w E Ap we also have that w - " ( ~ - ' )E Ap,,l/p + l/p' = 1. Let r = pp' and set T , f ( x )= ( M ( ~ f l p ' / ~ l / P ) ( ~ ) ~ ( ~and ) l / PT J) (l x/ )P=' ( M ( ~ ~ p w ' ~ p ) ( x ) w ( x ) - By l ~ the p ) l remarks ~p. at the beginning of this section it follows that TI and T2 are admissible and by Proposition 5.4 there is a
240
IX. Ap Weights
nonnegative, locally integrable function 4 simultaneously in A, ( TI) and A,( T,). This means that T , ~ ( x s )c ~ ( x ) or , M ( 4 p ’ w - 1 / p ) ( x )S c ~ ” ’ ( xW) ( X ) - ’ / ~
(5.3)
and T,+(x) s c ~ ( x ) or , M(+”w’l”)(x)c c+”(x)w(x)””. (5.4) In other words, c$pwl/pand +p‘w-l/pare A, weights. Put now w1 = 4pw1/p, w, = 4 p ‘ w - 1 / pand note that since p p’( 1 - p ) = 0 we have w1w i P p=
+
4P+P’(l-P)w1/Pw-(l-P)/P
=
Remark 5.6. It goes without saying that by Theorem 5.5, Ap weights satisfy properties (i)-(iv) in Proposition 4.5. 6. Ap AND BMO
As both the Ap condition and the definition of BMO deal with the, averaging of functions it is natural to consider whether there is any connection between these concepts. Proposition 6.1. Assume w is a nonnegative, locally integrable function. Then In w E BMO if and only if there is 7 > 0 such that w ” E A,.
Proof. We show the necessity first; by the John-Nirenberg inequality there are constants 7 ) s k/llln wll* and c, independent of I, such that
By removing the absolute values in (6.1) we also have
Whence multiplying the
+
and - estimates in (6.2) it follows that
that is, w“ is in A*. Conversely, assume such an
7)
exists and note that
6. Ap and BMO
241
say. Since both summands are handled in a similar fashion we only do A. By Jensen's inequality
s (A2
constant of
wq),
and we have finished. H A similar statement applies to Ap, namely, Corollary 6.2. Assume w is a nonnegative, locally integrable function, and for some 7 > 0, w" E Ap, 1 s p < 00. Then In w E BMO, Proof. If p s 2, then also w" E A 2 ,and the conclusion follows by Proposi, < 2, and again tion 6.1. If, on the other hand, p > 2, then w - " ' ( ~ - ' ) E A p rp' by Proposition 6.1 In( w - q ' ( p - l ) ) E BMO.
Proposition 6.3. Assume w is a nonnegative, locally integrable function. Then w E Ap if and only if
with c independent of I. As the proof should be obvious by now we omit it. Note however that by Jensen's inequality each factor in (6.4) is at least 1 and consequently the membership of w in Ap is equivalent to two separate conditions, to wit
and
An interesting application of these results is to evaluating the distance from BMO to L", more precisely, an estimate of the expression infgGL-ll4 gll* ,4 E BMO.What is relevant here is the quantity 7(4) defined as follows. Let 7 > 0, verify
IX. A, Weights
242
and put 7( 4 )= sup{7: (6.5) holds}. Two properties of this quantity are readily verified, namely, by the John-Nirenberg inequality 7( 4 )2 c/ )I4 (I*, c > 0, and ~ ( -4g ) = ~ ( 4for ) each bounded function g. We then have
Theorem 6.4 (Garnett-Jones). stants cl, c2 such that
There are absolute (dimensional) con-
c1/71(4) s infgsd14 - gll,
=Z
c2/7(4).
(6.6)
Proof. The left inequality in (6.6) follows at once from the comments preceding the statement of the theorem and we say no more. Next let 4 E BMO and pick 7 so that 7 ( 4 ) / 2< 7 < ~ ( 4 )by; Proposition 6.1 eT4 E A2 and consequently by Theorem 5.4 there are Al weights wl, w2 such that eT4(y) = wl(y)/ w2(y), or
74b)= In W l b )
- In
W2(Y).
(6.7)
Now since on account of Proposition 3.3 Mw,(x)" E A l , 0 < E < 1, with Al constant independent of wl, by Proposition 6.1 it follows that llln wlII* 6 absolute constant, and similarly for w2. Furthermore, since wl(y) zs Mw,(y) s cwl(y) a.e., the function g,(y) = ln(w,(y)/Mw,(y)) E L", and similarly for g 2 ( y ) = ln(w2(y)/Mw2(y)). Thus rewriting (6.7) as (In Mwdy) - In Mw2(y)) + (ln(w1(y)/Mwdy)) - 1n(w2(y)/Mw2(y)) = b ( y ) + g(y),sayweseeatoncethat4(y) = b ( y ) / +~ g ( ~ ) / r lIlg/71lms , 00, I(4 - g / 7 (I* s absolute const/ 7 s c/ 7(4 ) . Therefore the right inequality in (6.6) also holds and the proof is complete.
7. AN EXTRAPOLATION RESULT
This section is devoted to an important extrapolation property the A, weights verify; first we need some definitions and preliminary results. Definition 7.1. We say that the pair (w, v ) of nonnegative, locally integrable functions w, v satisfies the A, condition, 1 < p < 00, and we write (w,v ) E A,, if for all open cubes I and a constant c independent of I,
The infimum over the c's on the right-hand side of (7.1) is called the A, constant of (w, u ) , and a statement involving the pair (w, v ) is said to be independent in A, if it only depends on the A, constant of the pair, rather
7. An Extrapolation Result
243
than on the particular functions involved. Similarly, we say that ( w , u ) E A , provided that
where c is independent of I; the statement independent in A, has the obvious meaning. An example of a result independent in Ap is the following: let d p ( y ) = - w ( y ) dy, d v ( y ) = ~ ( y ) - ” ( ~ -dy, ’ ) v regular and doubling, then the weak type estimate (7.3) holds provided ( w , u ) E A p ,with the constant c in (7.3) independent in Ap. We do the case p > 1; first observe that, by Holder’s inequality and (7.1), for f b 0 we have
where c is the Ap constant of ( w , u ) . Whence by (7.4) it follows that, if
fi > A, then also L
Let now K be an arbitrary compact subset of { M f > A}, by the estimate (7.9, and as in (1.6) we see that p ( K ) s ch-’ J R n f ( y ) ’ d v ( y )with c independent in A, and we have finished. Another result of interest to us is Proposition 7.2. Suppose 0 < 7 s 1, 1 < p < 00 and w E A p . Let g L”;”(R”) and consider G ( y ) = ( M ( g ’ / ” w ) ( y ) / w ( y ) ) ” .Then
(i) [IGl[L$’”ScIlgllL$/” and, (ii) (gw, G w ) E A”+p(l-”).
E
IX. A,, Weights
244
Furthermore, both the constant c in (i) and the Al)+p(l-l)) constant of the pair (gw, G w ) are independent in A,,. Proof. Statement (i) has essentially been proved in Section 5. As for (ii), let q = 7 +p(1 - 7 ) and note that q - 1 = ( p - 1)(1 - 7 ) 3 0, or q 3 1. If 7 = 1, then also q = 1and since G = M ( g w ) /w we have that (gw, G w ) E A , with A, constant 1. Let then 0 < 7 < 1, on account of (7.1) we must show that
x
(i
~ I ( M ( g ' i n w ) ( y ) / w ( y ) ) - q l o l w ( y ) - ' / qd-y)"' '
s c (7.6)
for a constant c independent of I and independent in Ap.Now, by Holder's ~ its conjugate 1/1 - 7 we see at once that inequality with indices l / and
I,
Also since for each y in I M ( g ' / " w ) ( y )3 (1/1I1) g ( x ) ' / " w ( x )dx and q - 1 = ( p - 1)(1- v), the other integral in (7.6) is dominated by
Whence, by multiplying (7.7) and (7.8), we get that the left-hand side of (7.6) is bounded by
s (Ap constant of w)'-?, thus (ii) holds and we are done.
Remark 7.3. Proposition 7.2 may be restated as follows: assume 1 S po < p and w E Ap; then to each nonnegative function g in L F / P ~ " ( R "there ) corresponds a function G 3 g such that 11 GI1L',pIp~)' s c 11 g I( J ~ ' , / ~ O " and
245
7. An Extrapolation Result
(gw, G w ) E A,, with both c and the A, constant of the pair (gw, G w ) independent in Ap.Actually a stronger result holds, namely, Proposition 7.4. Assume 1 s po < p , w E A,,;then to each nonnegative func) may assign G 2 g such that IIGllLF/po)'d tion g in L ~ / P o " ( R "we c 11 g 11 L:Ipo" and G w E A,, with both c and the A, constant of Gw independent in Ap.
Proof. We proceed by induction. Let go = g and put g , in place of G in Remark 7.3. Here, g , verifies the estimate llg, 11 L$'po" S cllg 11 L$IPo", and by (7.3) the inequality
J
Ape {Mf>A)g o ( y ) w ( y )dv
k J R j f ( y ) l p o g l ( y ) w ( y )dy
holds for each f in Lp/po", A > 0 with constants c, k independent in A p . We can use g1 in place of go and so on; in general, given g,, we obtain gj+, 3 gj such that ~ ~ g , + , ~ ~ Lcllg,IILypo"d yP~)'~ ci+lllgollLypo" and the estimate
J
g j ( y ) w ( y ) dv
{Mf>At
k
J
If(y)lpogj+l(y)w(y)dv
(7.9)
(7.10)
R"
holds for every f in Lp/po",A > 0, with constants c and k independent in A,,. Now put G ( y ) ='C:o ( c + l)-jgj(y), where c is the constant in (7.9); since ( c + l ) - ' ~ ~ g j ~ ~ L $d~(mc /) c' + lyllgllLp'po" the series defining G converges in LjlpIPo)' and we readily see that G 2 g and IIGllLF'po)'d ( c + l ) ~ ~ g ~ ~ LMultiplying ~ ~ p o ) ' . each inequality (7.10) by ( c + l ) - j and summing over j we also get
i.e., the Hardy-Littlewood maximal function maps L2w(R") into wk-L2w(R")with norm independent in Ap. We may then invoke Theorem 2.1 part (iii) to infer that actually G w E A, with A, constant independent in Ap. Proposition 7.4 has a counterpart for the case p < po as well, namely, Proposition 7.5. Assume 1 < p < po and w E A p ;then to each nonnegative we may assign G 3 g such that 11 GI1Lf'po-p' s function g in LP,/'Po-P'(R") c ~ ~ g ~ ~ L $ ( pand o-pG ) - ' w E A,, with both c and the A , constant of G - ' w independent in Ap.
IX. Ap Weights
246
hf. We dualize Proposition 7.4; our assumptions are equivalent to 1 < p6 < p', u = w - ~ ' ( ~ - 'E) Ap,,and Ap.constant of 0 = Ap constant of w. Let d v ( x ) = u ( x ) dx.Then by Proposition 7.4 we conclude that to each nonnegative h in L$"lP;'(R") there corresponds H 2 h such that [ ~ H ~ ~ L s~ p ' ~ p ~ ) ' c ~ ~ h ~ ~ and L ~ Hu ' ~ Ep A:, ~ ) with ' c and the A,; constant of Hu independent in Ap.But (p'/p&)'= (po - l)p/( po - p ) so that h E L?'lp;)' is an equivalent -') Also Hu E Aio if and only if way of saying h P o - l ~ - ( p ~ - P ) l (EPLpVl(po-p). ( Hu )- I / ( P ; - ' ) = ( H P o - l W - ( P o - P ) / ( P - I ) ) - l W A . Thus given a nonnegative function g in L',/(po-P), we put g = hpo-lW-Ppp,-p)/(p-l) with h in L$P'/PO', obtain the corresponding function H from Proposition 7.4 and then define G = HPo-1 w-(Po-P)/(P-I).
We are now ready to prove the extrapolation theorem alluded to at the beginning of this section.
Theorem 7.6 (Rubio de Francia). Assume T is a sublinear operator which verifies the following property: there is a p o , 1 s po < 00, such that for every w E Am, d p ( x ) = w ( x ) dx,
IIrfllL2
cllfII~2,
(7.11)
where c is independent off and independent in Am. Then for every p with 1 < p < 00, and every w in Ap, T also satisfies the inequality
II
m.::
CllfII.:,
(7.12)
where c is independent o f f and independent in Ap. h f . We consider two cases; first suppose 1 s po < p , and let w E Ap and
f~ L',. As is readily seen
where the sup is taken over nonnegative g in L',"po" with IlgllLF'po"S 1. Fix such a function g and assign to it the function G 5 g of Proposition 7.4. Then by (7.11) and the properties of G given in that proposition we see that the integrals in (7.13) involving g are dominated by
where the constant c is independent of g and independent in Ap. Thus (7.12) holds and the discussion of this case is complete. Next suppose that
8. Notes; Further Results and Problems
247
1 < p < po and again let w E Ap and f E L:. Put g ( x ) = IlfI14Polf(x)lPo-P where f ( x ) # 0, and g ( x ) = 0 otherwise. Now note that
I
If(x)l”og(x)-’ d P ( X ) = llfll2;
(7.14)
{f+-O)
= 1. We are then in a position to invoke Proposition 7.5 and Ilg(lL;/(po-p) and obtain a function G 2 g with the properties given there. Observe that
where the constant c is independent in Ap.
8. NOTES; FURTHER RESULTS AND PROBLEMS
As expected, weighted inequalities are important in considering weighted mean convergence of orthogonal series, since the error terms can almost always be majorized by some version of the Hardy-Littlewood maximal function. In this context see Rosenblum [ 19621 and Muckenhoupt [ 19721. They are also important in the pointwise convergence of Fourier series as well: let s * ( f ;x ) = sup,Js,,(f; x)l, then Ils*(f)llL;S cllfllL;, 1 < p < 00, provided w E Ap (cf. Hunt and Young [1974]). Ap weights and their basic properties have been studied extensively; for instance, Feff erman and Muckenhoupt [ 19741 showed there are doubling measures which are not Ap weights for any p 3 1, and Stromberg [1979b] constructed examples to show that aside from the obvious implications, conditions such as doubling and reverse Holder as well as others we discuss in this section are independent of each other. The proof of Muckenhoupt’s Theorem 4.1 we present here is essentially due to Sawyer [19821 and Jawerth [ 19841 and it does not rely on the (difficult) implication “Ap implies Ap--.” as did the original proof. Sawyer’s idea is somehow related to the notion of Carleson measure which will be discussed in Chapter XV. The reader will note, however, that once the elements for the proof are set up, it very much looks like a Calder6n-Zygmund decomposition argument, especially relation (4.1). In fact Christ and R. Fefferman [1983] have shown that this is precisely the
ZX. Ap Weights
248
case; we prefer to give the more abstract proof since it applies to the very general context considered by Jawerth. The proof of the Jones Ap decomposition theorem given here is due to Rubio de Francia [1984]and that of Rubio de Francia’s extrapolation theorem is due to Garcia-Cuerva [1983].
Further Results and Problems
8.1 Suppose a nonnegative Borel measure p is doubling. Show that (R”
‘p(y) =
Let p be a regular, nonnegative Borel measure and let f~ L,(R”). Show that for some constant c, independent o f f , and A > 0, A p ( { M J > A}) S ~ j { ~ ~ , ~ J fd(p y( y)) I. (Hint: For a fixed A let 0 = { M J > A} and put f = fxo + f ~ ~ n = , ~f1+ f 2 , say. If we can show that 6 E { MJ1 > A}, then by Theorem 1.1 we are done; but this is easy since to each x in 6 there corresponds an open cube Z containing x such that (l/p(Z))j I l f ( y ) I d p ( y ) > A, which in turn implies that inf, M J > A and f = fl on I.) 8.3 The proof of Theorem 1.1 relied on a careful selection procedure of cubes out of an arbitrary family; results of this type are known as “covering lemmas.” That proof may also be obtained by making use of the following covering lemma, due to Wiener: Let E be a Borel measurable subset of R ” which is covered by the union of a family of open cubes {I,} of bounded sidelength; then from this family we can select a disjoint subsequence { r j } so that p ( E ) d c C p ( r j ) , where c is a constant that depends only on the doubling constant of p. Prove this lemma. (Hint: Choose I1essentially as large as possible, i.e., sidelength ZI2 i sup,(sidelength I,), discard any cube which intersects Zl,and so on.) 8.4 Maximal results, in turn, imply covering results; the following is an example: Assume that for a nonnegative, Borel measure p and some 1 < p < co the mappingf+ MJverifies IIMJllL: s cllfllL;, c independent o f f ; then given any finite collection of open cubes {Z,} it is possible to select a sequence { r j } so that (i) Z,) S c,p(U Zi) (that is the Zi’s cover Z,) and (ii) jR.(C x1,(y))”’d p ( y ) S c 2 p ( u r j ) (that is a good portion of the overlap of the Is’s is small when measured in L$( R ” )norm, l/p + l/p‘ = 1);the constants c l , c, depend only on the norm of the maximal operator and on p. (Hint: Since the Is’sare finitely many they may be ordered and we choose the first cube as I ] . For 1, we choose the first I, among the , n Zl)S $p( Z,). For Z3we choose the first I , remaining cubes so that p ( I among the cubes listed after Z2 so that p(Zp n (IIu I,)) S ip(Z,) and so on. Note that if an Z , was not selected then we have p(Z, n 4)) > ?p(Z,) 8.2
u
p(u
(u
8. Notes; Further Results and Problems
249
consequently p(uI s ) S p ( { M p ( x u I j> ) $1)c C ~ ~ I ) X U I ~=I I L ; c , p ( u $ ) and (i) holds. Next observe that if Ek = I k \ U j < k l j , then E L(&) 3
and
&.&(Ik).
We define now a linear operator T: L;(R") + LP,(R")as follows:
Clearly I Tf(x)l e M d ( x ) .Moreover, the adjoint T* :L$( R " ) + L$( R " ) can be explicitly written as
and consequently
and by taking p' norms we get (ii). this technique of proof is known as "linearization" and since at no point did we use the fact that the Is's were cubes the reader is invited to state and prove a general result in this direction. The substitute result for the case when the maximal operator is of weak-type ( 1 , l ) should also be considered. The proof above is from C6rdoba's work [ 19761 and the weak-type result was done independently by C6rdoba [ 19761 and Hayes [1976].) 8.5 Under very general conditions Corollary 1.3 admits the following converse (we only discuss the unweighted version here): a collection B = {B} of open, bounded subsets of R" is said to be a translation invariant Buseman-Feller, or B-F, basis if for each x in R" there is a subfamily B ( x ) of B such that (i) if B E B ( x ) , then x E B, (ii) each B ( x ) contains sets of arbitrarily small diameter; (iii) B ( x ) = x + B(0). Suppose that B differentiates L ( R " ) , that, is limlBI+o, B,sacx,(l/lBI) j B f ( y )dy = f ( x ) a.e.; then the mapping . f ( x ) M a f ( x ) = supBGB(x)(l/lBI)I B l f ( y ) l dy is Of weak-type ( 1 , l ) . (Hint: Suppose not and proceed exactly as in Chapter IV; the result is from de GuzmLn and Welland [1971].) 8.6 Although Ap is both necessary and sufficient for the ( p , p ) type and weak-type of the maximal operator, the same is not true for the restricted d p ( x ) S c p ( E ) ,all A > 0 and measurweak-type: the inequality A P able subsets E c R", is equivalent to the existence of a positive constant K such that for all cubes I and Lebesgue measurable E E I, lEl/lIl c K ( p ( E ) / p ( I ) ) ' / " (Hint: . Since M x E ( x )2 ( ~ E ~ / ~ I ~the ) ~condition ,(x) is necessary. Conversely, we readily see that M x E ( x ) K ( M d E ( x ) ) ' / P and that p is doubling. The result is Kerman's and the proof appears in Kerman and Torchinsky [ 19821.)
I{MxE,A)
IX. Ap Weights
250 8.7
For a nonnegative, locally integrable function w and an open cube = (tlf2)1/2 where tl = sup{t > O:I{x E I : w ( x ) s t}l s 1Z1/2} and tz = inf{t > 0: I{x E I : w ( x ) > t}l s lIl/2}. Show that for any real num. ber a we have m,a(Z) = (mw(Z))"and ( w " ) a~$ ( m w ( Z ) ) aFurthermore, we say that w satisfies condition A if w1 S cmw(Z),for a constant c independent of I. It is clear that, if 0 < a d 1 and w satisfies condition A, then ( w a ) , (m,( I ) ) a ,all I. Moreover, Apweights have the following interpretation in terms of the condition A : w E Ap if and only if w and w - ' / ( ~ - ' ) both satisfy condition A , 1 < p < 00. (Hint: One direction of the last statement is trivial since (m,( Z))-l/(p-l) = mW-m-1)(J). The other follows from the inequalities W r 2 imw(I ) = $(r n , - m - l ) ( and ( W - l ' ( p - l ) ) r 2 $ ( m w ( I ) ) - l / ( p - lThese ). results and those in the next remark are due to Stromberg and appear in Stromberg and Torchinsky [19801; they should be compared with Proposition 6.3.) 8.8 Let 1 < p , r < 00; then w E Ap n R H , if and only if w r and W - I ' ( ~ - ' ) satisfy condition A. (The statement is equivalent to w" satisfies condition A for all - l / ( p - 1) d a < r.) 8.9 Assume that a nonnegative function w verifies l{y E I ; w ( y ) < W r / A k } l s cvklIl,all I, where c is independent of Z and 0 < 7 < 1 < A < 00. Show that there is a p > 1 such that w E Ap. (Hint: Note that
Z put m , ( I )
-
and choose p so large that 7A1'(p-1) < 1.) We say that w verifies Muckenhoupt's A , condition, and write w E A,, if to each 0 < E < 1 there corresponds 0 < S < 1 so that for measur) IEl < 8111. By Proposiable subsets E of I we have p ( E ) < & p ( I whenever tion 4.5(iii) each A, weight is an A, weight. Show that the converse is also true: if w E A , there is 1 < p < 00 so that w E Ap. (Hint: It suffices to show that for appropriate A and 7 the assumptions of 8.9 hold; A = 8 and 7 = (1 - d / 2 ) , where d is the 6 corresponding to E = $ will do in the one dimensional case, the n dimensional case requires minor adjustments. To see this fix Z and k, let E = {x E I : w ( x ) < ~ 1 / 8 ~and } , observe that p ( E ) < P(I)/< ~~ p ( 1 ) / 2 implies [El < (1 - d)lIl. Now since almost every x E E is a Lebesgue point of xE and Lebesgue measure is regular we may assume 8.10
8. Notes; Further Results and Problems
251
that E is compact and each point of E is a Lebesgue point of ,yE. To each x in E we may assign an open interval I, centered at x such that 11, n Z n El = (1 - d ) ( I , n ZI (this is possible since for I, large I, contains I and IEl < (1 - d)lIl and for I, small I, c I and 11, n . E ~ / ~+ I , l), ~ I, G cZ, c independent of x and I. Let S = UxGE I,, since E is compact we may assume that S is finite and choose I , as an I, in S of largest length. Then after 11,. . . ,I k have been chosen let s k be the family of the remaining 1,’s so that x fZ 4 and let I k + l be a largest interval in s k . Observe that each y in 4 belongs to, at most, two of the 4 ’ s and put El = Uj(4n I) E Z. Then ~ ( ~ 61 Cj1 d d y ) G 2 C j J E n i j n i d p ( y ) (since IE n IJ n 1 = implies p ( 4 n I ) 6 2 p ( n (I - d)l4 n 1 ~ 4 n I ) ) s 4jEuijuid p ( y ) (since each y belongs to at most two of the 4 ’ s ) “41, d p ( y ) s 4 ~ ( I ) / 8 ~ . How about the Lebesgue measure of El? Well,
u
uj”=l
IIjnr
’
or lE.11,31El/~. Now, if k 3 2, it is possible to start with p ( E , ) < P ( E ) / ~ ~ - ’ and repeat the above argument with E replaced by El. This gives E2 c I, p(E2) 6 and lE21 > IE1/q2; repeating the process k times we are done. This result is Muckenhoupt’s [1974] and insures that P A , = Uisp<mAp.) 8.11 We say that a nonnegative, doubling, Bore1 measure p is comparable to a (similar) measure v if there exist constants E, 6 E ( 0 , l ) such that whenever E is a measurable subset of a cube I, v ( E ) / v ( I )< S implies p ( E ) / p ( I )< E. The following four conditions are equivalent: (i) v ( E ) / v ( I )s c ( p ( E ) / p ( I ) ) ”for all E c I, with c and 7) > 0 independent of E and I, (ii) v is comparable to p, (iii) p is comparable to v, (iv) d v ( x ) = w ( x ) d p ( x ) where w E R H l + ” ( d p ) i.e., ,
for every cube I. Moreover, comparability is an equivalence relation. (The proof uses ideas similar to the ones discussed in this chapter, (iii) implies (iv) is the hardest implication. This observation is from CoifmanFefferman’s work [ 19741.) 8.12 A nonnegative weight w E A, if and only if
IX. A, Weights
252
I,
( ~ i n t : Since limp+m((l/lIl) w(x)-'/(p-')dx)p-' = e(('/lZl)J, In('/w(x)) h) the assertion here is that A, is obtained as the limiting A, condition, as p -* 00, much like A, is obtained as the limiting condition as p + 1. The proof is computational, and the necessity follows at once from 8.11 and Jensen's inequality. As for the sufficiency, let A denote the sup over I of the expression in question, A < 00. Then for each interval I and disjoint subsets of positive measure E, F of I, E u F = I, we have In A 2 In( w , ) ( I E l / l W n W ) E - (IFI/lIl)(ln W ) F , where (In W ) E = (1/lEl) In w ( x ) dx, and similarly for (In w ) ~Since . by Jensen's inequality (In w ) S~In( w E )we also have that
JE
Putting t = IFl/lIl and T = p ( F ) / p ( I ) we finally get 1nA 2 (1 - t ) ln((1 - t)/(l - 7))+ t ln(t/T); elementary considerations obtain now that for a constant c, which depends only on A, T S 1 - e-'/('-'o) = T~ < 1 provided that t < to, in other words w E A,. The proof is from HruSEev's paper [ 19841.) 8.13 There is yet another way of writing the A, condition, namely, the S, condition
Jz
M(Xrw'-P')(X)PW(X)dx s c
J,
w(x)'-"' dx,
all
I,
with c independent of I. (Hint: It is not hard to see that S, implies A,. Conversely, if w E S, and x E I c Io, then by A, we readily see that
=S CM,( X I o w-') (x)"".
Since for each f with s u p p f r I. and x in I. we have M f ( x ) = s ~ P ~ ~ ~ = ~j J~f(( ~ )/dlI ~I itI, follows ) M(XroW'-P')(X)P c ~ , ( x ~ , , ~ - ' ) ( x ) ~ ' , and S, follows from Corollary 1.2. This proof is due to Hunt and Kurtz and Neugebauer [1983]; an indirect proof follows from 8.14.) Assume that 1 < p S 8.14 (Sawyer's Two Weight Maximal Theorem) q < 00, and that u, w are nonnegative, locally integrable functions in R". Then the following two conditions are equivalent: (i) ((,. M f ( x ) % ( x ) dx)'/¶d C ( ~ , . ~ ~ ( X ) ~ ~ dx)'IP, U ( X ) with c independent of J: (ii) I, M ( ~ r u ' - P ' ) ( ~ dx ) 4s ~ ( ~ ) U(X)'-"' d x ) ¶ I P < 00, all I, with c independent of I.
.(Iz
8. Notes; Further Results and Problems
253
The proof of this interesting result is due to Sawyer [19821, and it follows along the lines of Theorem 4.1. An identical result holds for M replaced by the maximal function M,,of fractional order introduced in Chapter VI. 8.15 With the same notation and assumptions as in 7.14 of Chapter 111, 7.29 of Chapter IV, and 6.17 of chapter VIII, prove the following: For every real sequence {ik}, -1 < rk < 1, with r; < a,the infinite product (1 + rkfk(nkX)) converges for almost every x to a function w(x) in A, for 1 < p c 00. Moreover, w E Lp(T), p < co as well. (Hint: the convergence of the product is equivalent to that of the sum s(x) = CT='=,rkfk(ng); also for some constant c, Iln w(x) - s(x)l< c. The convergence of s(x) follows by classical arguments. Observe now that for some constants cl, c2 > 0, cles(x)c w(x) < c2es(x).This result is from Meyer's work [1979].) 8.16 Under the hypothesis of 8.15 and if p*(x) = SUPN,O~N(X), pN(x) = n;="=,l+ rkfk(ng)), denotes the maximal Riesz product, then p* E A,. (Hint: First, there is a constant c > 0 so that (1/27r) JTp*(x)dx s c whenever CT=, r i c E 2 (this follows since pN(x) S cesN@)and s* E BMO). Let I c T, if II( > 27r/n0 the estimate holds trivially sincep*(x) 5 1 - ro > 0 and (l/lIl) J r p * ( x )dx s (no/2r) JTp*(x) dx. Suppose next that 27r/nN+, < 111 < 2 r / n N and let x = center of I; then there exists a constant c > 0 such that p j ( t ) c cpj(x), pj(x) c c p j ( t ) , whenever 0 c j S N and t E I. The constant c only depends on 1 r;. To see this we majorize Ilnpj(t) - lnpj(x)l by
n:='=,
i
Iln(1 + rkfk(nkt)- ln(1 + rkfk(n&)l k=O
i
d c
xlrklnzlx - fl
d c.
k=O
put Yk(t) = (1 + rN+lfN+l(nN+lf)+ * ' * + (1 + rN+kfN+k(nN+kt)) and note that p * ( t ) = sup(pl(t), * * ,PN(f), PN(f)Yl(f),. . . PN(t)Yk(f), ' * ') suP(pl(x), * . PN(X), PN(X)Yl(f), . . * , PN(X)'&(t), .) c sup(a, by*( t ) ) , where y*( t ) = SUpkal yk( t ) . Similarly, p*( t ) 3 c1 sup(a, by*( t ) ) 3 c,a. We will be finished once we show that (l/lIl) I, y * ( t ) dt < c, but this is not hard. The result is Meyer's.) 8.17 Assume w is a nonnegative function defined in a cube I. which verifies
-
-
9
3
( jzi
~ ( x ) dx)" "
c c1
-
ijz
W(X)dx
for all subcubes I c I,, some p > 1 and some constant c, independent of I. Show that there is 7 > 0 so that also
IX. A, Weights
254
for p S r < p + q, all I c I, and c2 = c,,,,, but independent of I. (Suppose I = [0,1], w ( x ) , d x = 1 and put EA = { x E I : w ( x ) > A}; the inequality follows at once from the estimate
I,
IEA
w ( x ) " dx d cAP-'
I,
w(x)dx,
A 3 1,
which in turn follows from an argument not unlike that of Theorem 3.5. This result, important both in applications and motivation, is due to Gehring [19731.) 8.18 For a locally integrable function 4 put p ( 4 ) = {infp: e b and e - @ belong to A,}. Note that p ( 4 ) can equal 00; also Holder's inequality . that p ( 4 ) = 00 and shows that ed E Ap whenever p > ~ ( 4 ) Suppose E BMO and p ( 4 ) - 1 = E ( + ) , where E ( + ) = show that inf{s > 0: supr(l/lIl)l{x E I : I+(x) > A}( =s e-"lE} whenever A > A. = AO(&, 4). (Hint: We must have p ( 4 ) =s2 (if p ( 4 ) > 2, then also e f b E A2, and this cannot be); then the A, estimates for e+' yield that
+
which in turn implies ~ ( 4=s ) p ( 4 ) - 1 since q ( 4 )= I/&(+). Conversely, for p - 1 > E ( + ) , again by the fact that q ( 4 )= l/&(+), we have that
and when p - 1 < 1, Holder's inequality shows that both eb and e-' The result is from Garnett-Jones [1978].) 8.19 Verify the following statements.
E A,.
(i) ( w , u ) E Ap if and only if (u'-~',w'-,') A,., 1 < p < 00. (ii) if (w, U ) E A,, O < S < 1 and ( q - l ) / ( p - 1) = S, then ( d ,u s ) E Aq. (iii) If (w, u ) E A,, 0 < S < 1, and d p s ( x ) = w s ( x )dx, d v 6 ( x ) = d ( x ) dx,then IIMfllL;, d ~ l l f l cl = ~ ~C6.p. (Hint: Since ( w ' , u s ) E A, and p > S ( p - 1) + 1 = q, we get that ({Mf > A}) s cllfll t:, and (iii) follows by the Marcinkiewin interpolation theorem. These observations and the next three res-ults are from Neugebauer's work [19831.) 8.20 Assume that
IlM-fIlL:
and IIMfllL!;-p. s CllfllLP;,-p,; then there are nonnegative functions w l , w2 such that w ( x ) l / p M w l ( x ) c l u ( x ) ' / p w l ( x )an ; identical inequality holdsfor w2, and W ( X ) ' / ~ U ( X )=' ~ ~ ' W ~ ( X ) W ~ ( X ) (Hint: ~ - ~ . Cf. Theorem 5.5 and consider T f x ) = w ( x ) ' / p M ( l f l u - ' / p ) ( x ) + u ( x ) ~ " p s M ( ~ f l w ' ~ p s ) (where x ) ' ~ ss = p / p ' . )
ClIflL:
255
8. Notes; Further Results and Problems
Let (w, u ) E Ap and 0 < 6 < 1. Then there exists a nonnegative function u = ug such that clw'(x) c u ( x ) s C ~ U ( X ) ~and u E Ap. (Hint: Choose 0 < E, 7) < 1, 6 = ET. From 8.19 (iii) we know that
8.21
~ v where ~ Mwj ~ ~ s c ~' ~ ( u / w ) j" = ' ~1, , 2. Note Thus, by 8.20, ~ ~ /= w,w;-" that uE = wl(u / W ) ' / ~ W ; - ~3 C M W , ( M W ~ )3' - cwI ~ w:-"( u/ w ) ~ ( ~ -=~ C) W' €~ and thus clws c (MW,)"(MW,)""-~'S c2us. Put now u= (Mw,)yMw2)""-p'. 1 8.22 Let ( w , u ) be a pair of nonnegative functions, then there exists u E A, with c1w ( x ) C u ( x )C c2u(x),if and only ifthere is T > 1such that (w', uT) E Ap.(Hint: Since u, u'-~'satisfy a reverse Holder inequality, there is T > 1 such that u' E A, and ( w T ,uT) E Ap. As for the converse, use 8.21 with 6 = l/T.) 8.23 Suppose that w is a nonnegative function and show that ( w , M w ) E Ap, 1 < p < 00. In particular M f ( x ) " w ( x )dx c c I R n l f ( x ) I p M w ( xdx ) in the same range of p's. (Hint: The proof is reminiscent of Proposition 7.2 (ii). If wI = 0 there is nothing to show. Otherwise, note that inf,Mw 5 w I ; thus ( w , M w ) E Ap with Ap constant 1. Consequently, if u ( x ) = M w ( x ) ,the maximal operator maps L: into wk-LP,for 1 < p < 00 and by interpolation also into Lc for the same p's. The result concerning the integral inequality was originally proved by Fefferman and Stein [1971].) 8.24 Given a nonnegative function w and 1 < p < 00, the following conditions are equivalent.
I,.
I,.
(i) There is a nonnegative, finite a.e. function u such that M f ( x ) " w ( x )dx S c J,nlf(x)l"v(x) dx, c independent of J: (ii) J,n w ( x ) / ( l + 1 ~ 1 dx~ A)(x)g(x) w ( x ) dx,
for some g 3 0 with IlglJLJp/po)s = 1. Associate with g a function G as in Proposition 7.4 and apply the weak-type assumption. If, on the other hand, 1 < p < p o , use Proposition 7.5 instead. The result is Garcia-Cuerva's [ 19831.)
257
8. Notes; Further Results and Problems 8.28
w
E
A nonnegative function w is said to satisfy the AP,¶condition, or AP,¶,if
where c is independent of I. The infimum over the c's above is called the AP,¶constant of w and a statement is said to be independent in AP,¶if it only depends on the Ap,4constant of the weights involved. Show that, if T is a sublinear operator which verifies
for some pair ( p o , qo), 1 < po s qo < 00, and all w E Ah,%, where c is independent in Apo,40, then it also satisfies the same inequality for any other pair ( p , q), 1 < p s q < 00 with l / p - l / q = l/po - l/qo for every w E AP,¶ with norm independent in AP,¶.(The proof, which is similar to that of Theorem 7.6, is in Harboure-Macias-Segovia [ 1984b1). 8.29 Let Mf,(x)
1
= sup -{)f(Y)l I
lIIl-7
dY,
where 0 < 77 < 1,
and the sup is taken over all open cubes I containing x. Then, if 0 < l / q = l / p - 77 < 00 and w E AP,¶,there exists a constant c, independent off and independent in AP,¶,such that
(I,"
M,f(x)'w(x)¶dx)liq
l/P
-
c( j R " I f ~ X ~ l P W ~ X ~ ~ d X )
Is the converse true? (Hint: The proof follows by a combination of the ideas discussed in this chapter; as an illustration we do the (easier) weak-type result. Assume that ~ , n l f ( x ) l p w ( x ) p d=x1 and note that by Holder's inequality and AP,¶we get that
the weak-type estimate follows without much difficulty from this. (Cf. the remarks in 8.14).)
258
IX. Ap Weights
8.30 For 0 < l / q = l / p - (Y < 1 the condition Ap,q is necessary and sufficient for the mapping
to verify
(cf. Muckenhoupt-Wheeden [19741). Welland [19751 observed that one may prove this inequality by using Theorem 2.4 in Chapter VI and 8.29. Results similar to 8.24 in this context have been established by Rubio de Francia [ 19811 and Harboure-Macias-Segovia [ 1984al.) 8.31 Liifstrom [1983] has shown there exist no nontrivial translation invariant operators on LP,(R”),if d p ( x ) = w ( x ) dx and w belongs to a class of rapidly varying weight functions, including for instance w ( x ) = e*lxla, (Y
> 1.
CHAPTER
More about R"
1. DISTRIBUTIONS. FOURIER TRANSFORMS
As we saw in Chapter IX the transition from periodic functions to those defined in R" may be accomplished smoothly. In this chapter we sketch some of the basic properties of distributions, functions, and operators in Euclidean n-dimensional space which will be useful in what follows. Some of the results are straightforward extensions of the corresponding statements in T and some are not; we will be brief in all cases, though. For instance, the Calder6n-Zygmund decomposition is available at all levels now (given A > 0, partition R" into a countable grid of cubes I with 111 > r, where l l f l l r < A, and observe that the "old" Calder6n-Zygmund process at level A applies on each cube I since < A ) and the Riesz potential operators ) L ~ ( R "for ) IJ(X) = JRnf(y)/lx- yl"-" dy, o < a < n, map L ~ ( R "into l / q = l / p - a / n (same proof as for periodic case). On the other hand, other results require some adjustment. For instance, to discuss the notion of distribution we introduce the Schwartz class Y'(R") as follows. Given an n-tuple of nonnegative integers a = ( a 1 ,. . . , a,) with length la1 = a , + . - * + a , , and x = (x,,..., xn) in R", we put x a = x;ll - x z m and define the differential operator D" = d"l/dx;ll * . ax>. The space Y ' ( R n ) consists of those C"(R") functions +(x) (i.e., all partial derivatives D " ~ ( xexist ) and are continuous) such that
InI
-
SUPlXP~"4(X)l= c , , p ( 4 ) < 03
(1.1)
X
for all multi-indices a, p. Y ( R " )contains the space C:(R") consisting of those C" functions with compact support, but e-IXl2 E Y(R")\CF(R").We say that a sequence {&} c Y ( R " ) converges to 0 in 9,and we write limk,, & = O(Y)if lirnk+" ca,P(&) = 0 for all multi-indices a, p. A linear 259
X . More about R"
260
operation T is said to be continuous in Y provided that limk+coT(&) = o(9) whenever limk+oo& = o(9). The reader may verify that all usual operations, such as addition, multiplication by a polynomial, and diff erentiation are continuous in 9. An essential operation which we hope to show to, be continuous in Y is the Fourier transformation. The Fourier transform f of an integrable function f is defined by the absolutely convergent integral r
f(t)= J R" f(x)e-"'&dx,
5 E R",
where x * 5 denotes the usual scalar prodyt x l & + * * * + xn&. Somz proper( ~0,) ~ ties of the Fourier transform, such as llfllm s Ilflll and l i m ~ + o o ~ f = are readily verified. Since, by the Lebe;gue dominated co_nvergenceth%orem, and t"f(6) = it follows that for f in Y ( R " ) ,O a f ( [ )= (-i)l"'(x"f) (t), (i)'"'(O"f)A(t), we also see at on2e that the Fourier transformation is continuous in 9.Moreover, since f E LJR") for f in 9,we consider the possibility of expressing f in terms o f f by means of a Fourier inversion formula. In the process of establishing this fomAulawe need to have at hand a specific integrable function 4 so that 4 is also integrable. We construct an example as follows: if n = 1 we put 4(x) = e-lxl and observe that
and
As for arbitrary n note that the Fourier transform of ~ ( x = ) +(xl) * is S ( t ) = 2"/(1+ 6:) * * (1 + 5;) and
*
- 4(x,)
r
(1.5)
We are now in a position to show Proposition 1.1. Suppose f
holds.
E
Y ( R " ) .Then the Fourier inversion formula
26 1
1. Distributions. Fourier Transforms
Proof. For 71 as above, and since all double integrals involved are absolutely convergent, it readily follows that
"
=
Thus,Aby replacing T(E*)
(y) =
R"
(1.7)
f ( x + y ) ; i ( v ) dv.
~ ( by t )T ( E ~ ) ,E > 0,
E-"~(Y/E),
J
J
in (1.7) and observing that from (1.7) it follows at once that
T ( E t ) ~ t ) e i pd*t R"
=
J
R"
f ( x + E y ) ~ ( ydu. )
(1.8)
We now let E + 0 in (1.8) and by the Lebesgue dominated convergence theorem we get
and the proof is complete.
Remark 1.2. A simple variant of t,he proof above shows that, properly interpreted, (1.6) still holds true iff is merely integrable.
+ ( x ) $ ( x ) dx
(4 *
i,hm i ( t > 4 < S ) =
=
(27r)-"
and (4i,hT(t)= (2+"(C$
* i)(S>.
Proof. The immediate verification of these properties is left to the reader. Next we consider tempered distributions.
X . More about R"
262
Definition 1.4. A linear functional F on Y ( R " )is said to be continuous if limk+mF ( + k ) = 0 whenever limk+m4 k = O ( 9 ) . The collection of all continuous linear functionals on Y(R") is called the space Y'(R") of tempered distributions.
As usual L P ( R " c ) Y ' ( R " ) ,1 s p C 00, and the action of the functional f corresponding to the function f in L P ( R " )is given by the absolutely convergent integral f(4)= I , . f ( x ) + ( x ) dx, 4 E Y ( R " ) . Finite Bore1 measures p, as well as functions of tempered growth, i.e., those functions f such that f ( x ) = 0((1 + 1 ~ 1 ) ~ for ) some integer k also generate distributions in the obvious way. Tempered distributions F are also infinitely differentiable in Y",and D"F is defined by D"F(4) = (-l)'"'F(Da4), all 4 in Y ( R " ) Clearly, . differentiation is continuous in Y ' ( R " )(we say that Fk-f O ( Y ) if Fk(4) 0 for each 4 in Y ( R " ) ,differentiation preserves this property). As in Chapter I, we may prove -f
Proposition 1.5. A linear functional L on Y ( R " )is a tempered distribution if and only if there exist a constant c > 0 and integers k, m,which depend only on L, such that IL(4)I C CC,,~c",~(+), /a[ k, IPI m. We also have (cf. Theorem 4.12 in Chapter I) Proposition 1.6. A tempered distribution F is supported at xo E R" if and only if F is a linear combination of the Dirac 6 concentrated at x,, i.e., the distribution 6 ( 4 )= 4(x,,) and a finite number of its derivatives. As -for the Fourier transformation we have Definition 1.7. A tempered distribution F !as a well-defined distributiona! Fourier transform # given by fi(4)= F ( + ) , a! 4 in Y ( R " ) .Moreover, F verifies the following inversion formula: if 4(x) = 4(-x) and F ( 4 ) = 2 F(I$), then F = (2.rr)"fi We also have the following important property Proposition 1.8. Suppose f E L2(R").Then the (distritutional) Fourier transfomf off coincides with an L2(R")function and llfll2 = (2.rr)"/'llfll2. Furthermore, Proposition 1.3 holds for 4, in L2.
+
Proof. By the Detnition 1.7,*HOlder's inequality and Proposition 1.3 y e have If(+)I = ItI I I ~ I I ~ I I = ~ I( I2~ . r r ) n / 2 ~ ~ f ~ ~all2 ~4~in4 ~9~. n2 ,u s f E L2(R")_and IIfll2 (2.rr)n'211f((2.Also replacing f by f we see that (2.rr)"llfllz = (2.rr)"llfllz = llZll2 (2.rr)n~211Z112 (2.rr)"llfll2, and the proof is complete. W
2. Translation Invariant Operators. Multipliers
263
Corollary 1.9. Assume f E L P ( R " ) , 1 < p < 2. Then the Fourier t r a y form f of f is in L p ' ( R " ) , l / p + l / p ' = 1, it is given by f = 1imN+? ~lxl,Nf(x) e-&'*dx, where the limit is taken in the LP'(R")norm and Ilfll,, s Ilfl ,. Proposition 1.10. Suppose that the distribution F has compact support. Then ~ ( 5 =) F(e-"'*) is a c"(R") function. Definition 1.11. For F E Y ' ( R " ) , 4 E Y ( R " ) we define the convolution F * 4 at x as the C"(R") function F ( T , ~ )where , T, denotes translation by x in R". By Proposition 1.5, F * 4(x) has tempered growth and its Fourier transform is the distribution $, where the product is understood in 9".A similar definition holds for the convolution of distributions, one of which has compact support. 2. TRANSLATION INVARIANT OPERATORS. MULTIPLIERS A bounded linear operator T from L P ( R " )into L 4 ( R " ) is said to be TTT~ for all x in R". Unlike for the periodic translation invariant if T ~ = case, such operators do not always exist. Proposition 2.1 (Hormander). If T is a bounded, translation invariant operator from L p ( R " )into L 4 ( R " )and q < p < CO, then T = 0. Proof. First note that limlxl+&+ ~ X f l l ,= 21/pIlfllp;the proof of this is not hard once we write f = u + v, where u is compactly supported and llull, is small, and observe that for 1x1 sufficiently large the support of u and that of T,U are disjoint. We show next that the smallest constant c for which 11 TfII, < cllfll, is 0. By the linearity and translation invariance of T we get IITf+ ~,Tf11~ = IIT(f+ ~ ~ f lC 1 c1l ~ l f + ~ , f l l , , and consequently, letting 1x1 + co it follows that 11 Tfll,s 21/p-1/q cllfll,, where the factor 21/p-1'q< 1. By repeating this argument we see that indeed the smallest such constant is 0 and the proof is complete. W
Next we would like to discuss the relation between translation invariant and convolution operators; this requires the following result, which is a variant of the well-known Sobolev lemma. Proposition 2.2. Suppose f E L P ( R " ) 1, s p s CO, and f has distributional derivatives of order S n + 1 which coincide with LP(R")functions. Then f equals a continuous function a.e. and there is a constant c = c,,,, independent off and x such that If(x)l s c Clal,n+lllDafllp, x E R".
X . More about R"
264
Proof. First, note that for 6 in R", (1 + I6l)"+' s c &+n+ll[UI for some constant c independent of 6. Suppose first that p = 1 and observe that
c I(~"f)A(5)1 + /&'""' c 11
Ij.cs>l sz 4 1 + 161)-'"+1'
Irrl 1 and choose 4 in C:(R") so that +(x) = 1 if 1x1 s 1 and 0 if 1x1 > 2. Then fi$ verifies our assumptions for p = 1, and by the above proof it coincides a.e. with a continuous function h such that
c , , , ~ , ~ " ~ ~ ( x ) D " ~it ~ ( x ) , Since by Leibnitz's rule D"(f+)(x) = Ca1+a2=o( follows that the right-hand side of (2.2) is bounded by c maxIa2I+III D"24 [IoD ClullSn+l11 DU1fllp,with a constant c which depends only on n and p. Moreover, since 4 = 1 in 1x1 s 1, h actually coincides with f in that neighborhood of the origin, and the same argument applied to f~&, x E R", gives the general result.
We may now prove
Theorem 2.3. Suppose T is a linear, bounded operator from L P ( R " )into L 9 ( R " ) , 1 p G q 00, which commutes with translations. Then there exists a unique tempered distribution F such that Tc#J(x)= F * +(x), for each 4 in Y ( R " ) . Proof. Since T is translation invariant, Tc$(x) has derivatives of all orders which coincide with L 4 ( R " )functions and Da( T 4 ) = T(D"c$),all a;this observation follows at once since the difference quotients of 4 converge to the corresponding partial derivatives of 4 in Lp, and, consequently, T of the quotients converges to T of the derivatives in L9. By Lemma 2.2, it now follows that Tc$ coincides with a continuous function, after correction on a set of measure 0, which verifies IT4(0)l
c
c
(olJ 0}, we pass to discuss the generalization of the Hilbert transform to R". First, we find the Poisson kernel P(x, t) in R:+', that is, a function which verifies (i) ( ( $ / a t 2 ) + A)P(x, t ) = 0, (ii) I,. P(x, t ) dx = 1, t > 0, and (iii) lim,,o P ( x , t ) = 6 (in 9'). Taking the FIourier translform in the space x-variables, by (i] we note that (a2/at2)P(5,t ) - ([I'P([, t ) = 0 and by (iii) that limr+oP ( $ t ) = 1. Thus solving the differential equation we have that P ( & t ) = cl(5)e-'lf' + ~ ~ ( 5 ) ewhere " ~ ~by, the tempered nature of P we have c2(() = 0
X . More about R"
268
and by the above observations cl(.$) = 1. In other words, @(&,t ) = e-r'gland c
P(x, t ) = (2T)-"
J
- 451 k . 5 Rn
45.
(3.9)
It is not hard to find the explicit expressions for P(x, t ) . Indeed, by (3.9) we see that it is of the form t - " 4 ( l x l / t ) and from (i) above it follows that 4 satisfies the differential equation
+ s'>c"'(s) + ( 2 ( n + l ) s 2 + ( n - l))+'(s) + sn(n - l ) + ( s ) = 0, s > 0.
s(l
Thus 4 ( s ) = (1 + s2)--("+')/' and P(x, t) = P r ( x )= cnt-"l/(l+ (Ixl/t)2)("+')/2, (x, t ) E R:+', where c, = (27r)-" j R n e-lC1dt. By passing to polar coordinates, i.e., by putting d t = rn-l dr d f , where dt' is the surface area element on I; = (6' E R": 16'1 = l}, we get that c, = (27r)-"(n - 1) ! w,, w, = surface area of X. Polar coordinates also make it possible to compute w,; indeed observe that e-IXl2dx
=
(IR
ePs2ds)' =
I
re-r2
dx' dr
= T.
109)
Thus lXI2
IR"
dx = T n / 2
= w n I [o,m)
-
(">
e - r 2 d r = - r - w,, 2 2
and our computation is complete. Now for f in LP(R"),1 d p < 00, u(x, t ) = f * P ( x ) is the unique harmonic function in R:+' such that lim,,,u(x, t ) =f(x) in Lp and s ~ p , , ~ j ~ ~ It)lp u (dx x ,= < 00. It is natural to consider whether u is a (first) component of a system of conjugate functions R:+'. Disregarding the question of convergence of the integrals involved, we note that
llfllz
and by (3.7) the other (possible) components of the system of conjugate functions satisfy
=
f * Qkb, t),
(3.10)
where Qk(x, t ) = (2T)-"c,,xk/(t2+ IX/~)("+')/~ denote the conjugate Poisson kernels, 1 d k d n. Since as is readily verified ( d / d t ) u + ( d / d x , ) u , + * + (d/dx,)u, = 0, the vector is indeed a system of conjugate functions. What
-
269
3. The Hilbert and Riesz Transform
are the limits of the &(x, t)'s as t + O? From (3.10) it is readily seen that they are the Riesz transforms Rk given by
It is not hard to see that the Riesz operators are bounded in L2(E"),since fromtherelation Qk(x,t ) = (Xk/f) 1 (cf. 6.33 in Chapter VII). We may iterate (4.15) with p k = r k and 8 k = 1 - 1/2k7and, taking limits as k + m, the desired conclusion follows at once. Next we use the local boundedness of u to estimate up and In u.
X . More about R”
276
Proposition 4.12. Suppose u is a nonnegative solution to L and 2 1 c CR. Then there are constants c, m > 0 such that for any -1 c p s 1, we have X ) provided f S t < T S 1. suptl up C ( c / ( T - t ) l I l l / ” ) m,1 U ( X ) ~ W ( dx Proof. We may suppose inf, u > 0 (the essential inf that is), and put 4 ( x ) = $ ( X ) ’ U ( X ) ~ - ~ , s # 0, 1, where $ E C p ( 2 1 ) . Since by Proposition 4.11, u is also bounded above in 21, 4 E H and consequently 2 ( u , 4) = 0. This gives the inequality
(4.16)
-
as long as s # 0, 1. Given now any pair of values .C t < T c 1 we choose a function 7 E C : ( T I ) such that 7 = 1 on t I , 0 C 77 S 1 and I V v ( x ) l C 2/lZ11/”(T - t ) . Theorem 4.8 applied to ~ ( x ) u ( x ) ’ / ’ and combined with (4.11) and (4.16) gives
(
u (x ) “ ~ ”w ( x ) dx
In other words, if r = 4 / 2 > 1,
)”‘
E
= 11 - l/sl, then for any s # 0 , l we have
277
4. Sobolev and Poincare' Inequalities
The Moser iteration technique calls for repeated use of this inequality for a sequence of values of s, E, t. A choice that works in this case is tk = ;(I + z - ~ ) ,sk = sr k and E& = ill - 1/skl, where in order to avoid sk coming too close to 1 we take s of the form fr"(r + 1) for some integer u. Concerning In u we have
Proposition 4.13. Suppose u is a nonnegative solution to L, 21 c a. Then there is a constant c such that A ~ ( {ExI : Iln u ( x ) - (In u ) [ l > A } ) c cp(1)/111'/''.
(4.18)
Proof. We may assume sup u > 0. Let U = In u, by Chebychev's inequality it suffices to prove that ( , I U ( x ) - UIl d p ( x ) s c p ( I ) / ~ I ~ ' Now ~ " . since U E H we may invoke PoincarC's inequality to get
J I v u( x >~'
s c~(I)
dv(x).
(4.19)
I
To estimate the right-hand side of (4.19), we introduce the function 4 ( x ) = t,b2(x)/u(x),where $ E C;(21), = 1 on I, 0 s t,b s 1, and IVt,b(x)(s c/lZl"". Since (by replacing u by u + 6 if necessary and then letting 6 += 0) we may assume that u ( x ) 3 6 > 0 and by Proposition 4.11 u is bounded on the support of $, the function 4 E H and 2 ( u , 4) = 0. From this one easily gets that ( 4 ( x ) 2 a (U ( x ) , U ( x ) )dx c 4 ( a ( 4 ( x ) , c$(x)) dx and consequently
(4.20)
Whence combining (4.19) and (4.20) our conclusion follows. W We still need one more result essentially due to Moser [1971], the proof of which is also achieved by an iteration technique and is left for the reader to attempt.
Proposition 4.14. Assume {E(t ) } is an increasing family of measurable sets,
f s t c 1. Let g be a nonnegative, bounded function on E ( 1) which satisfies the following conditions.
X . More about R n
278
(i) There exist constants c, rn > 0 such that SUP^(^) gp d : T - I)-"' g ( x ) ' d p ( x ) , holds for 0 < p < 1, f 6 t 6 T s 1. (ii) There is a constant c such that A p ( { x E E(1): In g ( x ) > A}) d c.
IE(=)
Then there exists a constant k, depending only on the constants c, rn of (i) ( ~ k ~ ~ ) and (ii) above such that S U ~ ~g d We are now ready to state and prove Harnack's inequality
Theorem 4.15. Assume u is a nonnegative solution to L. Then for any compact subset K of R there exists a constant c = cK such that sup u s c inf u. K
(4.21)
K
P m f . We may, and do, assume that u ( x ) 2 6 > 0 a.e. in R. Our conclusion will follow once we show that to each x in R there corresponds an open cube I containing x and so that sup u d c inf u, J
(4.22)
J
where the constant c in (4.21) depends only on I. Z in fact is a cube centered at x so that 41 c a, 1211 C 1 and p(21) C 1. Put A = (In U ) 2 J and consider the functions U l ( x )= e - " u ( x ) and U 2 ( x )= e"u(x). We claim both of these functions satisfy (i) and (ii) in Proposition 4.14 for E ( t ) = 2tZ, f s t d 1. Indeed, U, , U2are bounded because u is bounded above and below. Now, according to Proposition 4.12 we have
for 0 < p < 1 and f s t < T s 1 and (i) holds for U,; a similar argument shows it also holds for V 2 . As for condition (ii), we note that since {x E 21: In U l ( x )> s} c {x E 21: Iln u ( x ) - A1 > s}; it holds for U , on account of Proposition 4.13, and similarly for U2.Whence by Proposition 4.14, and with a constant k which may depend on Z but not on u, we get sup U ] , I
sup J
u, d s
(4.23)
and, consequently, since U , ( x )U 2 ( x )= 1, the inequalities in (4.23) can be rewritten as k-' d U , ( x )d k,all x E Z, and sup U, S k2 inf, U,.The desired conclusion follows now by multiplying through by e". It is well known that this result has important consequences, such as the strong maximum principle (a solution to L which attains its maximum or
279
4. Sobolev and Poincare' Inequalities
minimum in
is constant) and local Holder continuity, max u - mjn u s r
1
1,
1/q
u ( x ) ¶d p ( x ) ) ,
where q depends on the Ap condition of w and I is well inside 0;for further details the reader should consult Gilbarg and Trudiner [ 19771. The exposition of the material presented here follows that of Harboure [19841 and is related to previous work of Fabes, Kenig and Serapioni [1982] and Chanillo and Wheeden [ 19851.
CHAPTER
Calderon-Zygmund Singular Integral Operators
1. THE BENEDEK-CALDERON-PANZONE PRINCIPLE In this chapter we extend the continuity results we discussed for the Riesz transforms and more general odd kernels in Chapter X to arbitrary Calderbn-Zygmund singular integral kernels. We begin by proving a general principle which summarizes much of what we had to say concerning those kernels.
Theorem 1.1 (Benedek-Calderbn-Panzone). Suppose A is a sublinear R") into measurable functions which satisfies the followoperation from ing two conditions: (i) A is of weak-type (r, r), 1 < r < a.More precisely, for f in L'(R") and A > 0,
Cr(
Arl{lAA
'A l l =sc;llfllL
where c, is independent off and A. (ii) Let f e L ( R " ) , s u p p f ~B ( x o , R ) = { x E R": Ix - xol < R } , je(.+R)f(~)dx = 0. Then there are constants 1 < c,, c3, independent o f f and B ( x o , R) such that
I
R"\B(xo,czR)
IAf(x)l dx
c3llfll1.
Then A also is an operator of weak-type (1,l); in other words, there is a constant c4 = c4( cl, c,, c, ,A), independent off E C;( R") and A > 0, such that Al{lAA
'A l l 280
C4llflll.
28 1
1. The Benedek- Calderbn- Panzone Principle
Proof. For f in C,"(R") and A > 0, let f = g + b denote the CalderbnZygmund decomposition o f f at level A (cf. Theorem 3.1 in Chapter IV). In particular,
Also by 7.2 in Chapter IV, limN,mll~JNlbj - bll' = 0. Now since by the sublinearity of A we have that IAf(x)ls IAg(x)l + IAb(x)l, it easily follows that
l{lAA
'All s l{IAg/> Wl1 + I{IAbl > WlI.
(1.1)
The measure of the set involving g in (2.1) is readily estimated since by c ~ l l g lS l~ Chebychev's inequality we have (A/2)'({lAgl > A/2)1 cc;Ar-'llfll1 and consequently Al(lAg1 > A/2)( S 2'ccillfl11, where c is a dimensional constant. The term involving b is a bit more delicate. First, observe that
IAb(x)l S CIAbj(x)l
a.e.
(1.2)
i
Indeed, on account of tile subadditivity of A we have
N
=
+ CIAbj(x)I,
~ N ( x )
(1.3)
j=1
say. Now since l i m N + m ~-~CJEl b billr = 0 and A is of weak-type (r, r ) , it readily follows that +N + 0 in measure, as N + 00, and consequently there is a subsequence Nk+ 00 so that 4 N k (+~0) a.e. as Nk+ 00. Using this sequence in (1.3) gives (1.2) at once. We need one last observation of a geometric nature before we proceed with the proof. Let Bi denote the ball concentric with 4, with diameter Bj = c2 diameter Zj, and put R, = U Bi. Then lRll G CIBil S c ClfJls (c/A)llfll1, where c is a dimensional constant. With this remark out of the way we return to estimate Al{IAbJ> A/2}1 S AIR1l + A({x E R"\R,: IAb(x)l> A/2}1= I + J, say. We just noted that
282
XI. Caldero'n-Zygmund Singular Integral Operators
AIRll S cllflll, and I is of the right order. As for J, observe that by Chebychev's inequality and (1.2). J s2
j
IAb(x)l dx s 2
R"\O,
j c
IAbj(x)l dx
R"\ni j
I,
Moreover, since each bj verifies b ( x ) dx = 0, by (ii) each summand above is bounded by c3 5,.1b(x)1dx 6 cc3 srJlf(x)ldx, and J C C ~ ( ( ~ which ( ( ~ , is also of the right order. Remark 1.2. Clearly, the same proof applies to L:(R"), the space of compactly supported, bounded functions on R".
Before we proceed with the applications, we discuss the case r covered by Theorem 1.1.
= a, not
2. A THEOREM OF ZO
We begin by proving
Proposition 2.1. Suppose A is a sublinear operator which satisfies the following three conditions: (i) A is of type (a,a) More . precisely, IIAfllms clllf)lm, where c1 is independent of J: (ii) Same as condition (ii) in Theorem 1.1. (iii) For every sequence (4) of pairwise disjoint open cubes and every integrable function h supported in 4 and such that I, h ( x ) dx = 0, allj,
u
IA~(x)I
CIA(hXrj)(x)I
ax-
i
Then A is also of weak-type ( 1 , l ) .
Proof. For f in L ( R " ) and A > 0, consider the Calder6n-Zygmund decomposition off at level h, f = g + b. Since llgIlm 2"A, \Ag(x)\ c12"h and {IAgl > t } = 0 whenever t 2 c12"A. Therefore, {IAJ > ~ ~ 2 " +E !~ h } { ( A b (> c,2A}, and by (ii) and (iii) the measure of this set is estimated exactly as in Theorem 1.1. An interesting application of this result is to maximal operators of the form
283
2. A Theorem of Zo'
Our assumptions on the k,'s are those which will insure that (i), (ii), and (iii) of Proposition 2.1 hold. For (i) to hold it is readily seen that we must have
y z [R"lka(Y)l
dY s c,
(2.1)
i.e., { k a } a E Ais bounded in L ( R " ) .As for (iii) note that since
then also =0
N+w
for each
a
E
A.
Therefore there is a sequence N , + w such that
and consequently Ih * ka(x)I s CKhXr,) * ka(x)I
a-e.
i
which gives (iii) at once. Finally we consider (ii), this will require a new assumption on the ka's. We must estimate
I=
suplk, xo-xlZ=c2R
* f(x)l dx.
(2.2)
P
Now since jB(xo,R)f(~) dy = 0 we have k, * A x ) = jl.o-ylsR(k(~ -Y )k ( X - XO))f(Y) dv and SUPlk *f(x)l
[
SUPlkU(X- Y ) - kP(X - XO)lIf(Y)I dY.
.x--ylSR
(2.3)
01
Therefore, by Tonelli's theorem, putting (2.3) into (2.2) gives Is
j
Ixo-Y~S R
If(Y)I(
j
(r,-xlac2R
SUPlk,(X
- v) - k ( x - X0)l dx 4Y (2.4)
)
and (ii) will hold provided the innermost integral in (2.4) is finite. More precisely, we have Theorem 2.2
(26). Suppose { k a } a E verifies A the following two conditions
(i) JRnlkol<x>l dx (ii) ~lxlaczl,,l sup,[k,(x - y ) - k,(x)l dx s c3 where 1 < c2, c3 are constants independent of y E R".
XI. Calderbn- Zygmund Singular Integral Operators
284
Then the mapping f ( x )
T f ( x ) = sup,)k, * f ( x ) l is of weak-type (1,l).
Corollary 2.3. Suppose a nonnegative, integrable function 6 verifies IV+(x)l s c / I x l n f ' . Then if k E L ( R " ) and lk(x)I s 4 ( x ) , the mapping f ( x ) + SUP,,^/^^^ t-"k((x - y ) / t ) f ( y )d y ( is of weak-type (1,l).
Proof. Cf. 8.4 below. Corollary 2.4. Let k be as in Corollary 2.3 and suppose f E L ( R " ) . Then
Proof. Cf. Corollary 2.4 in Chapter IV.
3. CONVOLUTION OPERATORS How does Theorem 1.1 apply to convolution operators? Theorem 2.2 hints that the kernel k in question should verify the following conditions:
k
E
L(R")
(3.1)
and
By Young's convolution theorem, condition (3.1) alone implies thatf + k * f is bounded in L P ( R " ) ,1 d p C 00, with norm cllklll. But the kernels of interest to us, such as those corresponding to Riesz transforms, fail to be integrable in a neighborhood of the origin and at infinity. Aside from this they are locally integrable in R"\(O) and they satisfy (3.2), also known as Hormander's condition. Our strategy to deal with this more general situation consists of three steps, to wit: (1) Truncate k both at 0 and 03, obtain an integrable function, tr k say, and observe that the mapping f + tr k *f is well defined in L P ( R " ) , 1dpC0O.
(2)
Estimate Iltr k
*fIIp
s cllfllp, where c is independent of I(tr kill.
1 A l l s
Cllflll,
(5.7)
AI(1TY-I > A l l s cllfll1,
(5.8)
c independent of A, E,J; and
with c independent of A, f: Proof. Assumption (iv) insures that the proof given in Theorems 1.1 and 2.12 in Chapter V for the Hilbert transform also works in this case with minor adjustments. Another way to go about this is to observe that the
6. Maximal Caldero'n-Zygmund Singular Integral Operators
291
estimate (5.6) insures that hypothesis (ii) of Theorem 1.1 holds, and consequently that result also applies. (Also cf. Theorem 3.1 in Chapter XIII.) There still remains the question of the pointwise convergence of the truncated CZ singular integral operators. This requires the simultaneous control of T&,Nf;and is achieved in the next section with the consideration of the maximal singular integral operators. 6. MAXIMAL CALDERON-ZYGMUND SINGULAR INTEGRAL OPERATORS
For a CZ kernel k let
T* is called the maximal CZ singular integral operator (associated to k ) and our aim is to show that under appropriate conditions the statements analogous to Theorems 5.4 and 5.5 hold for T* as well. Lemma 6.1 (Cotlar). Suppose the CZ kernel k verifies properties (i) through (iv) of Definition 5.1 and Theorem 5.5. Then for 0 < 77 < 1 and Tf as in Theorem 5.4,
T*f(x) 4wI~fl")(x)"" + M f ( x ) ) , (6.2) where c = c" is independent off E C r ( R " ) and x. Proof. As indicated above, for compactly supported functions f the limit as N + m of TE,Nfcan be easily handled, so we may just consider Tef: Now given compactly supported functions f; g it is readily seen that there is a sequence cj + 0 so that lim T J w )
&,+O
=
Tf(w)
and
lim T,g(w) = g ( w )
EJ+0
(6.3)
exist simultaneously for almost every w in R". Fix x in R" and E > 0 and for a given f in C r ( R " )let g ( y ) = f ( y ) x e c x , , , ( ybe ) the restriction off to the ball centered at x of radius E. Next observe that TEf(x)=
I,
X-yI>&
( k ( x - u ) - k ( w - Y)lf(Y) dY
XI. Calderbn- Zygmund Singular Integral Operators
292
say. In (6.4) we have chosen w to be a point B(x, ~ / 4 where ) (6.3) holds. Clearly, by (the first limit in) (6.3) lim&J+o K = T f ( w ) .As for the J term, ) sj + 0, upon choosing E~ sufficiently small it follows since w E B ( x , ~ / 4 and = that B( w, E ~=) B(x, E),and, consequently, for those ~j’sf(y)~Rn\B(~&)(y) ~ ( ~ ) x R “ \ B ( ~ , & ) ( Y ) x ~ ( ~ , ~n~ ) u(sY )J. may be rewritten as -Ilw-yI,EJ k(w y ) g ( y )dy and by (the second limit in) (6.3) lim,J+oJ = - T ( ~ x B ( & ~ ) ) ( w ) . Since Z is independent of E ~by , letting E~ + 0 in (6.4), we obtain that for almost every w in B(x, .5/4) T&f(X)=
J;
(k(x - Y ) - k ( w - Y ) ) f ( Y ) dY
X-YI>E
- T ( f X ~ ( x , & ) ) ( wTf(W) ) =I say. Now, by property (iv) of k, it follows that
J
K,
(6.5)
where 4(x) is the radial, decreasing, integrable function (1 + IX~)-(“+~), 4,(x) = E - ” ~ ( x / E )and , by Proposition 2.3 in Chapter IV, (I1 S cMf(x). Whence by (6.5) we see that ~)I (6.6) IT&f(x)) cMf(x) + ) T ( ~ X B ( X , E ) ) (lTf(w)l for almost every w in B ( x , ~ / 4 ) .NOW,if TEf(x) = 0, then clearly (TEf(x)I is bounded by the right-hand side of (6.2). Otherwise, let 0 < A < ITJ(x)l, ) B and introduce El = { w E B: ITf(w)l> A / 3 } , E2 = put B(x, ~ / 4 = { w E B: IT(fxBCq&))(w)l> A / 3 } and E3 = 0 if cMf(x) S A / 3 and = B if cMf(x) > h / 3 ( c is the constant in (6.6)). From (6.6) it is clear that B = El u E2 u E,, and we pass now to estimate the measure of these sets. In the first place by Chebychev’s inequality
Similarly, and invoking Kolmogorov’s inequality 7.19 in Chapter IV and Theorem 5.5, we see that
and since lE21 is finite, we also have
6. Maximal Caldero'n-Zygmund Singular Integral Operators
293
As for E,, it either equals B, and in this case M f ( x ) / A > c > 1, or it is empty. In either case, P 3 1 6
lBlMf(x)/A.
Thus combining (6.7), (6.8) and (6.9) it follows that
IBI
=S cA-"
J i T f ( w ) l " d w + cA-'
I
lf(Y)ldY
B(x,E)
which, multiplying through by A / [ BI, in turn gives
+
s c(A'-"M((Tfl")(x) M f ( x ) ) .
(6.10)
+
Now, since from the inequality 0 < A G clA'-" c2 we readily get that A s max((2cl)'/", 2c2) s ( 2 ~ ~ ) "+" 2c2, by (6.10) we obtain A G c ( M ( ) T f ( " ) ( x ) ' / " M f ( x ) ) ,and (6.2) follows at once from this. H
+
Theorem 6.2. Assume the CZ kernel k verifies the assumption of Lemma 6.1. Then IIT*fllP
CllfllP,
1 A l l s
cllfll1,
A
> 0,
(6.12)
for a constant c independent of A and f: Proof. One way to prove (6.12) is to repeat the argument of Theorem 2.14 in Chapter V. Another way is to note that on account of Lemma 6.1, it is enough to show that for f in C?(R"), A I ~ A= I AI{M(ITfl")(x)> A'II
cllfIIi,
Now, by 8.2 in Chapter IX, we have that 10Als cA-"
A
> 0.
(6.13)
I, I Tf(x)l"dx
0 / { A ( f )> A}I 9 I{T*h > A / 2 } (s cApS. Since S is arbitrary we immediately get that / { A ( f )> A } / = 0 for each A > 0. Thus A ( f ) ( x )= a.e.; in other words lim sup,,, T E f ( x )= lim inf,,, T E f ( x )a.e. and the limit exists for each f in L p ( R " ) .Furthermore, since it coincides with T f ( x ) for f in C;(R"), a simple argument along the lines of Theorem 2.2 in Chapter IV shows that the same is true for an arbitrary f in L P ( R " )and we have finished. I 7. SINGULAR INTEGRAL OPERATORS IN L"(R")
As in the periodic case the class BMO arises as the image of L" under CZ singular integral operators. Theorem 7.1. Suppose the CZ kernel k verifies the assumptions of Theorem 5.5, let k , ( x ) = k ( x ) if 1x1 > E and 0 otherwise, and for f in L"(R") set
KEf(X) = j
y
x - Y ) - kd-Y)lf(Y) dY.
(7.1)
Then lime,, K E f ( x )exists a.e. in x and is a BMO function; more precisely, there is a constant c independent off such that
IIKfll*
cllfllm(7.2) Proof. Observe first that, on account of property (iv) of CZ kernels, the integral in (7.1)converges absolutely for each E and x ; in fact, it is precisely for this reason that the term - k , ( - y ) was introduced. Also note that iff is compactly supported, then K E f ( x )differs from T , f ( x ) by a constant. Moreover, well-known arguments by now show that limE,, K , f ( x ) exists in the L2 norm on each finite cube as well as pointwise a.e. in R".
8. Notes; Further Results and Problems
295
Next consider K , f ( x ) - K N f ( x ) = kc,N* f ( x ) , where by Lemma 5.2 the integrable function kE,Nis a CZ kernel with constants uniformly bounded, independently of E and N. It is not hard to see that IIkc,N*fll* s C I I ~ I \ ~ , with c independent of E, N, and f ; in fact, the proof of this estimate is similar to that of Theorem 3.1 in Chapter VIII and is therefore omitted. Thus there is a constant c, independent of E, N, and f, so that
Next observe that if cN = IRn(kN(-Y) - k , ( - y ) ) f ( y ) dy, then K N f ( x ) C N = I p ( I C N ( x - y ) - k N ( - y ) ) f ( y ) dy tends to 0 uniformly as N + 00. Moreover, since the inequality in (7.3) remains unchanged if we replace K N f by K N f - cN, we get, by first letting N + CO and then E + 0,
8. NOTES; FURTHER RESULTS AND PROBLEMS Because of the many applications to other branches of analysis and PDE’s, the Calder6n-Zygmund theory of singular integral operators plays a basic role in harmonic analysis and lies at the heart of much of the work being done in this area nowadays. Although other mathematicians, most notably Giraud and Mikhlin, obtained n-dimensional results, it was only through the techniques introduced by Calderbn and Zygmund that the complete picture began to emerge. The classic 1952 Actu Mathemuticu and 1956 American Journal of Mathematics papers make inspiring reading and, even though they are quite well understood by now, there is yet much to be learned about the precise meaning of the various conditions discussed there. For instance, only recently Calderbn-Zygmund [19791 showed that in the case of kernels of the form k ( x ) = n(x’)/lxl”,n homogeneous of degree 0, the Hormander condition (iii) *of Definition 5.1 is actually equivalent to the following one: For a proper rotation p of R “ about the origin put /PI = suplx’ - px’l. Then if w , ( t ) = sup^^^^^ I,)k(px’) - k(x’)l dx’, ~ ~ o ,wl( l l t ) / t dt < 00. Also Calder6n and Capri [ 19841 have shown that if T is as in Theorem 5.4 and f and Tf are both integrable, then limE-,ollTJ Tflll = 0. The versatility of the methods discussed in this chapter is apparent in the consideration of the so-called CZ operators, corresponding to “variable”
296
XI. Calderbn- Zygmund Singular Integral Operators
kernels (also cf. Calder6n and Zygmund [ 1978]), and oscillatory singular integrals. Further Results and Problems 8.1 De Guzmin [1981] observed that the method of rotations applies to the Hardy-Littlewood maximal operator as well. In other words, assume that the 1-dimensional maximal operator is bounded in Lp, 1 < p < 00, and prove that the same is true for the n-dimensional maximal operator. What can be said for the case p = l ? 8.2 Suppose k ( x ) = f2(xr)/\xl", homogeneous of degree 0 and I,n ( x ' ) dx' = 0, is a CZ kernel and show that
lnll/cos
$1
T
- i-sgn(cos 4)
2 where $ denotes the angle between [ and x'. (Hint: we have
In polar coordinates
say. I is easily computed; as for R note that
+I
[ E COSl&'l,l)
(cos r - 1) - + dr r
I
[I/=)
dr cos r-, r
and take the limit as E + 0. The fact that jx f2(x') dx' = 0 is crucial.) 8.3 Suppose the kernel k verifies (ii) in Definition 5.1 and the mappings T , f ( x ) = k, * f ( x ) are bounded in some L P ( R " ) 1, < p < 00, uniformly in E. Then for each 0 < E < N, IJEE(S)
(s - t )
Now interchange the order of integration in the expression above and observe that since E ( S ) is arbitrary, we may conclude that, if
then also l\g*11, S cllgll,. To pass to the n-dimensional statement it suffices now to invoke the method of rotations, i.e., to use an argument similar to Theorem 3.1 in Chapter X; this result is from Calder6n-Zygmund's work [1956]. R. Fefferman [1979] observed that even in the case when b is an arbitrary bounded function the following is true: suppose that for each r > 0 we are given a function a, on X in such a way that the family {a,} is uniformly in the Dini class; i.e., if w ( t ) = sup{la,(x') - a,(y')l: Ix' - y'( < t, r > O}, then w( t ) / t dt < a;and also I ,a , ( x ' ) dx' = 0. Let h ( x ) = fllxl(x')/Ixln,then 1lp.v.h *fl12 s ~ l l f l 1 ~ .If in addition the Dini condition is replaced by a Lipschitz condition of some positive order, then also 1k.v. h *f",S cllfllp, 1 < p < a.The case p = 2 amounts to showing that h is bounded, but things get complicated when p # 2 since in general nothing - y ) - h(x)l dx. The proof relies can be said about integrals like ~lxlP21,,llh(~ then on the complex method of interpolation. Namazi [1984] relaxed the Lipschitz condition and characterized those b's for which the corresponding mapping is bounded from L"(R") into BMO(R").Also Shi [1985] proved results for more general CZ operators.) 8.7 Suppose that is a nonnegative function, supp c (1x1 s l}, such ,.+ ( x ) dx = 1. Let E > 0 and put & ( x ) = E - " + ( x / E ) and define that I 6 , ( x ) = T + € ( x )- k E ( x ) ,where k is a CZ singular integral kernel and T is the CZ operator associated to k. Show there exists a constant c > 0 such that 116,ll, s c, uniformly in E. (Hint: Suppose first 1x1 3 2 s and observe that by the Lebesgue dominated convergence theorem T+&) = JlylSE k ( x - Y ) & ( Y ) dy. Consequently, = jl,,lSE(k(x- Y ) k ( x ) ) + , ( y ) d y and the appropriate estimate for j l X 1 2 2 B 1 8 Edx ( ~follows )I from Fubini's theorem and condition (iii) in Definition 5.1. On the other hand
+
I,
X1S2E
18,(X)l dx
+
I,
XIS28
I W € ( X ) I dx +
j
lkI dx
=
I + J,
E 4'+' 4"+', a , , ( t ) s 16.4-' if It f 6 4"'+', u , , ( t ) G 4-2J+mif 4' + 4,+' s It I s 4J+' - 4'"+', finally a.(, t ) = 0 if It1 < 4' - 4"+', q mt() s 16.4-J if It + 4'1 s 4"+'. Consequently, f R a',,,( t ) dt s 50.4-'+" = a ( j - m ) , which is precisely what we wanted to show. This result is from David's work
+
[1982].)
8.11 Motivated by 8.10 and the results in this section we propose the following definitions: we say that a possibly complex-valued, continuous function k ( x , y) defined for x # y is a Calderbn-Zygmund kernel provided the following two conditions are satisfied: (i) Ik(x, r)lc CIX - yl-". (ii) The distributional derivatives D"k(x, y ) , IaI = 1, coincide with -n-1 . locally bounded functions in x # y and verify ID"k(x, y)I S Ix - yl Associated to k we define the operator T by means of the formula T f ( x ) = Pev*I R n k ( x 7 y ) . f ( y )dy = lirn~;rOjI~-yI>~ k ( x , y)f(y) d y 7 f E C?(R"), and we say that T is a Calderbn-Zygmund operator provided T admits a continuous k(x, y ) f ( y ) dy extension to L2(R").As expected we denote T , f ( x ) = IIX-yl,E and T * f ( x ) = sup,) T,f(x)l. Prove that Calderbn-Zygmund operators are bounded on L P ( R " )1, < p < 00, and map L ( R " )into wk-L(R"). Also prove that the same result holds for 7'' and T * . (Hint: The proofs in Section 5 apply to this setting as well; the notation, as well as the formulation of the next four results, is due to Coifman and Meyer [1978].) 8.12 Show that the conclusion of 8.11 holds under the following assumptions on k in place of (i) and (ii) there: (iii) there exist a constant c and a number 0 < S c 1, such that, if Iy - yo[c r and ( x - yo/ 2 2r(x, y , yo E R", r > 0) we have Ik(x, y)I s cIx - yl-" and Ik(x, y ) - k ( x , yo)l, Ik(y, x ) k ( y o ,x)l s cr 6 Ix - yol-n-6. In fact, Yabuta [1985] has shown that the k suffices: Ik(x, y ) ( s cIx - yl-" following assumption on and lk(x, y ) - k ( x 7 yO))+)k(y, k(y07 x ) \ C ~ x - ~ o ~ - " w ( ~ ~ - ~ for all x, y, yo with 21y -yo[ < Ix -yo\, where w is a nonnegative, nondecreasing function with j(o,llw( t ) / t dt < a.
~ ~ / ~ x - ~
301
8. Notes; Further Results and Problems
8.13 Suppose T is a Calder6n-Zygmund operator which verifies the assumptions of either 8.11 or 8.12. Then for each function f in L P ( R " ) , 1 < p < CO, lim,,o T , f ( x ) exists and coincides a.e. with T f ( x ) defined as 4 k l l p -,0. follows: Tf = limit in Lp norm of TcPkwhere 4kE C r ( R " ) and 8.14 If T is a Calder6n-Zygmund operator a n d f e C r ( R " ) ,then for each x in R" we have T ' ( x ) ~ c l l T J J M ( l f l ~ ) ( x ) ' where '~, l cMf(xo). Since I is arbitrary our proof 4 ( y ) lnlyl dy < 00 is complete. A similar argument shows that, if IlylSEN instead, then the stronger inequality M & Tf(x) d cM#f(x) holds. These results, as well as related ones, are in Jawerth and Torchinsky's work [19851.) 8.17 Let k E Y ' ( R " ) have compact support and let 0 < 8 < 1 be given. Further, suppose that ,k coincides with a locallyAintegrable function away and from the origin, that k is a function and that Ik(S)l d A(l + 1~$1)-"' / ~ - y ) - k(x)) dx d c for lyl d 1. Then the convolution operator jlxl,21yl~-~,lk(~ T'= p.v. k *J;is bounded in L P ( R " ) ,1 < p < 00, and maps L"(R") into BMO(R"). (Hint: We may assume that k is integrable by replacing, if necessary, k by k * 4€,where 4 is a C,"(R")function with integral 1, and observing that the above conditions are satisfied by k * 4€,uniformly in E. We show first that T maps L" into BMO. Let I be a cube of dianeter 6, which we may assume is centered at the origin. Of the two cases, 6 S 1 or 6 > 1, we only do 6 d 1. Write f =f,+ f 2 , where f,= f in the ball 1x1 d 26'-', f 2 = f -4, and u1 = Tfl, u2 = Tfi. In terms of Fourier transforms n0/2 * GI([) = )5(-"e/2k([)1S1 f,([), where according to our assumptions $(t)I#'e/2 is bounded. Thus, by (the n-dimensional variant of) Theorem 2.1 in Chaptzr VI, u ^ , j s the FouGer transform of an Lp function with norm ~All~(S)~S~"""f1(S)ll2 cllfll12 cllf,l12, with l l p = 1/2 - 8/2. in other words Thus IrIul(x)pdx d cllf,ll; d CllfIlmNow let a, = k(-ylf,(y) dy. Since UZ(X) (l/lIl) f,IU1(X>1dx a, = jRn(k(X- y ) - k ( - y ) ) f , ( y ) dy, if 1x1 d 6 (which is the case if x E I), we get that IuAx) - a,( (Ilyl,21xll-elk(~- v) - k(-y)l dy)llfllm. Clearly T is bounded in Lz, and by Theorem 4.2 in Chapter VIII, T is also continuous in L P ( R " )for 2 < p < 03. The statement for 1 < p < 2 follows by duality. These operators were introduced by Fefferman [19701, who also proved that they are of weak-type ( 1 , l ) . The fact that they map L" continuously into BMO was proved by Fefferman and Stein [1972].)
CHAFTER
XI1 The Littlewood-Paley Theory
1. VECTOR-VALUED INEQUALITIES It is often possible to extend inequalities involving scalar valued functions to functions which take values in a Banach space and thus obtain not only a more general result but also one which can be applied to other situations. The purpose of this chapter is to take systematic advantage of this fact. We begin by giving two important examples, one concerning maximal functions and the other Calder6n-Zygmund singular integral operators. In each case , c: p < Q), and the applications will be discussed the Banach space is l P ( Z ) 1 later on.
Theorem 1.1. (Fefferman-Stein). Let f = (fi,. . . , f k , . . .) be a sequence of functions defined on R " and, corresponding to f, consider the sequence Mf = ( Mfl . . . ,Mfk, . . .) whose kth term is the Hardy-Littlewood maximal function Mfk offk. Then II II~fkIlI'llp CII Ilfkllrdlp7 1 < r7 P < Q), (1.1) where c = c,,, is independent of$ Also, and with a constant c = c,,~independent off,
NlMfkllI' > All cll llfklll,lllY 1 < r < Q). (1.2) Proof. To simplify notations put F ( x ) = Ilfk(x)lllr and m F ( x ) = IIMkf(x)I(,r.We consider separately the cases p < r, p = r, and p > r. When p = r (1.1) follows at once from the maximal theorem since
(1.3)
XII. The Littlewood -Paley Theory
304
where dJ E L'P/"'(R")and has norm S l . To estimate the integral in (1.4) we invoke 8.23 in Chapter IX and note that it is dominated by "
I f k ( x ) l ' ~ 4 ( xdx )
s
~ l l ~ l l ~ l l ~ sl lc~l l Fp l l/ i *r ~ ~
Therefore the right-hand side of (1.4) is bounded by cllFilp, which is precisely what we wanted to show. Finally since the remaining cases of (1.1) follow from (1.2) and the Marcinkiewicz interpolation theorem, we show (1.2). Consider the Calder6n-Zygmund decomposition of F at level A and in particular consider a family of disjoint, open cubes ( 4 )such that if R = 4, then ~ R ~ S ~ ~ F ~ ~ , / A , F ( x ) 6 A for x in R"\R and ( l / l 4 l ) IrJF ( x ) dx s 2"A, all j . Let now fk = g k + hk, where gk =f xXR" \Cl, hk and put G ( x ) = Ilgk(x)IIIr,rnG(x)= IIkfgk(x)l(lr andsimilarlyfor H and rnH. Since Mfk(x)s Mgk(x) + Mhk(X), all x and k, it suffices to show that rnG and rnH are in wk - L ( R " ) , with norm s c 11 FII 1 . This is immediate for m G since 11 G 11 zs cA 1) FII and by (1.3) ((rnG((, zs cllGllr; whence A'({rnG> All cA'-'llFll~. The estimate for H requires some work. In the first place let f k ( x )= Cj((l/141) j j , f k ( y )dy)xrJ(X)and P(x), rnF(x) as usual. Observe that supp F c a, and for x E 4, by Minkowski's inequality, we, have F ( x ) s (1/I4I) Ir,IIh(~)IIi~d~ = (~/IJI) IjJ F ( Y )d ~ 2 " 5 ~ - Thus IIFII: s CAW s c A ' - ' J J F I J ~ and as above we see that Al{F > A}l 6 cllFII,. Our proof will (2n4), then for all k and thus be complete once we show that, if 6 = with a constant c independent of k,
u
uj
kfhk(x)zs ckffk(x),
a.e. in
~"\6.
(1.5)
To show (1.5) fix a cube Z containing x and note that
where the sum is only extended over J = {those j ' s so that I n 4 # 0).In this case, since x E Z \ 6 c I \ 2 n 4 , by the geometry of the situation, it follows
305
1. Vector-Valued Inequalities that I j G 2nI. Therefore, the right-hand side of (1.6) does not exceed
Since I is arbitrary we conclude that for x E R"\fi, Mh,(x) s cMfk(x)and we are done. A similar result is true for CZ singular integral operators, the key observation being that a statement analogous to 8.23 in Chapter IX holds in this case as well. More precisely, we have
Theorem 1.2 (C6rdoba-Fefferman). Assume k is a CZ kernel which verifies the assumptions of Chapter XI. If T denotes the CZ singular integral operator associated to k and w is a nonnegative function so that w s is locally integrable for some s > 1, then
"I,[ for all f
E
~ f ( X ) l " W ( X ) dx
s c [R"lf(x)lpM(w s ) ( x ) l / dx s
L P ( R " ) 1, < p < co, and a constant c
=
(1.7)
cp,sindependent off:
Proof. We begin by pointing out a variant of Theorem 7.1 in Chapter XI, namely, TfX(x)s c M ( l j y ) ( x ) 1 / q 7 1 < q < co, (1.8) with c = cq independent off: To see this fix a cube I containing x and put f = fx21+ fXRn\2I = fl + f 2 , say. Then Tf = Tfl + Tf2and it suffices to show (1.8) with f l and f 2 in place off in the left-hand side of that inequality. In the first place, by Holder's inequality and Theorem 5.4 in Chapter XI,
and this estimate is of the right order. Also for y in I,
s cM(lj-y)(x)'/",
and consequently the estimate (1.8) holds.
XII. The Littlewood- Paley Theory
306
Now since by Proposition 3.3 in Chapter IX,M ( w S ) ( x ) ’ / ’ E A,, from (a simple variant of) Theorem 4.4 in Chapter X and (1.8) it follows that
.1
I,. == 1.“
I T f ( x ) l ” w ( x ) dx
1 and by Theorem 4.1 in Chapter IX we see that the righthand side of (1.9) is dominated by c ~ , . ~ f ( x ) ~ p M ( w ” ) ( xdx, ) ‘ ~and ” the proof is complete. 4 We are now in a position to prove Theorem 1.3 (C6rdoba-Fefferman). Let (4)be a sequence of CZ kernels with unformly bounded CZ constants and let {Ti} denote the sequence of CZ singular integral operators which correspond to the 4’s. Then
where c = c,.,
II Il~LllrIl,c cll II.6III~llP9 1 < r,P < 00% is independent off = (fl, . . . ,A,. . .).
(1.10)
Proof. Since a simple duality argument shows that the estimate (1.10) holds with indices p, r if and only if it holds with indices p’, r‘, l/p + l/p’ = l / r + l / r ’ = 1, we may assume that p 3 r. The case p = r is a simple sequence of Theorem 5.4 in Chapter XI.On the other hand, if p > r, then the left-hand side of (1.10) equals
where g E C , “ ( R “ )and ~ ~ g ~ S~ ~1. Now p , r by ~ rTheorem 1.2, and with 1 < s < ( p/ r)’ there,
IlfjIIV l l ~ l l ~ ~ l ~ l s ~ ~ ~ s l l ~ ~ CII II.6IIr 1; llglk,/rY,
sz CII
and we are done. We consider next a general result in the direction of Theorem 1.4 and some of its applications.
2. Vector-Valued Singular Integral Operators
307
2. VECTOR-VALUED SINGULAR INTEGRAL OPERATORS We begin by discussing some preliminary results; we intend to be brief. Given a separable Hilbert space H with inner product (,) and norm lhlH = Ihl = (h, h)"2, we say that a functionfdefined on R" and with values in H is measurable (or weakly measurable) if the scalar function ( f ( x ) ,h ) is Lebesgue measurable for every h in H. The class Lp(R",H ) consists of those measurable f with Ilfll, = ( j , . l f ( x ) l P d ~ ) < " a, ~ 1 p s 00, and similarly ~ ~ =f ess ~ sup,nlfl ~ m denotes the norm in L"(R", H). For a couple of separable Hilbert spaces HI, H2 let B( HI,H2) be the (Banach) space of bounded, linear operators T from HI into H2 endowed with the norm ITIB(H,,H2)= IT1 = ~ ~ ~ h e H , ( l ~ l H ~ / l h l H , ) . We say that a functionfon R" and with values in B( H1,H 2 )is measurable if f ( x ) h is an H2 valued, measurable function or each h in HI. In this case IflB(H,,H,, is also Lebesgue measurable and the spaces LP(R",B ( H , , H 2 ) ) may be defined exactly as above. The usual facts concerning operations of functions hold in this general setting as well. For instance, suppose that a function k defined on R" and with values in B ( H I ,H2) is integrable, and for f in LP(R",H,) put
Then the integral in (2.1), as an element in H 2 , converges weakly in H2 for d ~Furthermore, . almost every x, and l g ( X ) l H , IRntHx- Y)IB(H,,H,J~(Y)IH, llgllp s Il~lllllfll, 1 P Another important property concerns the Fourjer transformation. For f in L(R", H)we define its Fourier transform bzf(5) = jRne - i 2 T x ' v ( xdx. ) In this case is also H valued and clearly IIfllg, s llflll. Futhermore, if f E L(R",H ) n LZ(R",H),by means of an appropriate limiting process as was done in Chapter X, also f E L2(R",H) and Plancherel's identity is valid for these functions. This is readily seen by expressing the elements of the Hilbert space in terms of an orthonormal basis and then proceeding as in the scalar case. To further illustrate the fact that the results we need in this setting are simple extensions of the scalar case we prove the Marcinkiewicz interpolation theorem. Theorem 2.1. Let A be a sublinear operator defined on LF(R",HI), i.e., compactly supported, bounded H,-valued functions, with values in M(R", H2), i.e., the space of measurable, H2-valued functions. Suppose in Hl)AI{IAflH, > A)I 6 c l ~ ~ fand ~ ~ ArI{IAflH, l, > addition that forfin hm(R", A)[ s c ~ ~where ~ fc1 and ~ ~c, are ~ ,independent of A and f: Then for each
XII. The Littlewood - Paley Theory
308
1 < p < r, we have that Af E LP(Rn,H2) wheneverf E LP(R",H , ) and there is a constant c = cl,,,, independent off such that IIAfII, s cllfll,. Prmf. Let F ( x ) = (If(x)lH,)-'f(x) wheneverf(x) # 0 and 0 otherwise. For a scalar valued function g consider Bg(x) = jA(F(x)g)l,,. Clearly B is a sublinear mapping, simultaneously of weak-types ( 1 , l ) and (r, r), with norm s cl, c,, respectively. By the Marcinkiewicz interpolation theorem 4.1 in Chapter IV there is a constant c as indicated above so that llBgllpS cJJgJJ,, 1 < p < r. Upon setting g(x) = If(x)lH,our proof is complete. W An important result for our purposes is the following extension of Theorem 1.1 in Chapter XI,
Theorem 2.2. Suppose a linear operator A defined in LT( R", Hl) and with verifies values in M ( R " ,H2)
(9 ~ ' l { I A f l >~ 1 1 C l l l f l l r , some r > 1. (ii) Iff has support in B(x,,, R) and integral 0, then there are constants c2, c3 > 1 independent off so that
I
R " w x O .c2R )
IAf(x)l dx
C3llflll.
Then also Al{lAA > A}[ s cllfll, and by Theorem 2.1 also IIAfll, for 1 < p < r.
cllfll,
Since the proof is identical to that of the cited result, it is omitted. In the same vein we have Theorem 2.3. Let k be a function on R" whose values are bounded linear operators from H , to H,; we assume k to be measurable and integrable on compact sets. For f~ L,"(R", H , ) put
r If for some r > 1 and f in L'( R ",H,)the inequality 11 Tfll and
I
Ik(x - y ) - k(x)l dx
S ~ 2 ,
y
E
llfll ,holds,
c1
R",
lXl*2IYl
then Tf E LP(R",H 2 )for all 1 < p < 00 and 11 Tfll, s cllfll, where c depends on c,, c2 and p but is otherwise independent off:
3. The Littlewood-Paley g Function
309
The proof is identical to that of Theorem 3.1 in Chapter XI and is omitted. As for the vector valued singular integrals we have Definition 2.4. We say that a function k on R" whose values are bounded operators from H1to H2 is a vector-valued Calder6n-Zygmund integral kernel provided that (i) k is measurable and integrable on compact sets not containing the origin. (ii) For 0 < E < N, (jEclxl 0. (iii) f , n l @ ( x + y ) - +(x)l dx d clyl', y in R", some y > 0.
XU. The Littlewood -Paley Theory
310
Clearly any Schwartz function with vanishing integral verifies (i)-(iii). We also have PIoposition3.2. Suppose I)is a Littlewood-Paley function and 5 E R". Then
I$( 81H2s c*
Proof. We begin by showing that
1$(&>1
s c min(l~la'(n+l+a) 121-y), 9
(3.1)
where c is independent of 5. Since by identity (5.3) in Chapter XI $(2) = ~ j , . ( $ ( x )- $ ( x + y))e-"'*dx, y = .rr&/1@, by (iii) above it follows s c1,$1-'. Furthermore, since I) has vanishing immediately that l$(.$)I integral, we also have =I ,.$(x)(e-"'* - 1) dx, and consequently
$(e)
I$(S)l s 2 1,"I$(X)l min(lxl 121,1) dx s 2121
I
IxlI$(x)I dx + 2
IXlS?
I
IXI'?
II)(x)l dx
=I+J, say. Now since I C cl,$clq"+' and J C ccl j l x l > q l ~ I - ( n + adx ) = cq-O1, we obtain at once that I$(.$)I c c(I(lq"+' + q-"), and (3.1) follows upon minimizing with respect to 7. To complete the proof we invoke (3.1) and estimate
~$(.S>I'H, s c
j
min((tltl)a'(n+l+a),
dt ( t l 5 1 ) - " ) 2 y s c.
10.~)
Let now k ( x ) E L(C, L2(R+,d t l t ) ) be given by k ( x ) a = t - " $ ( x / t ) a = $,(x)a, where x E R", a is a complex scalar and $ is a Littlewood-Paley function. Corresponding to k we consider the singular integral operator
= lim E'O
J
- Y M Y ) dy.
(3.2)
Jx-yI>s
We want to show that T falls within the scope of Theorem 2.5 and thus obtain its Lp continuity, 1 < p < 00. To get a feeling for the situation we do the L2 case first.
3. The Littlewood - Paley g Function
311
Proposition 3.3. T is bounded from L2(R") into L2(R", L2(R , ,dt/ t ) ) .
Proof. Observe that for f in Lz(R") and on account of Tonelli's theorem, Plancherel's identity, and Proposition 3.2 we have
In fact a more precise result holds in the particular case $ is radial, namely Proposition 3.4. Suppose $ is a radial Littlewood-Paley function, then
Proof. As above we see that
4
and sinceA is also radial the innermost integral in (3.4) is readily seen to be I[o,co,I+Cl(t)IZ/tdt*rn For the other values of p we have Theorem 3.5. Suppose T is given by (3.2). Then
II Tfll,
cllfllp,
1F
We then set F ( x , t ) = f * & ( x ) and ITf(x)l = S,(F)(x), the Lusin (or area) function of F (with opening a). If we denote by T,(x) = { ( y , t ) E R:+': Ix - yl < at} the cone with vertex at x and opening a, it follows immediately that
As we did in case of the g function it may be readily seen that llSa(F)[[2 S cllfl12. Indeed, it suffices to observe that if y, denotes the characteristic function of the unit interval then
(4.3)
and the assertion follows at once from Proposition 3.3. In fact the above argument shows that under the normalization of (3.9), also IISJllz = Ilf112. Also an argument quite similar to that of Theorem 3.5 shows that k verifies (i)-(iv) in Definition 2.4 and consequently
IISJll,
ZS
cllfll,
1 < P < a,
(4.4)
4. Lusin Area Function and Littlewood-Paley gf Function
315
where c = cp,, is indepedent off: To prove the inequality opposite to (4.4) we proceed exactly as in Section 3, so we say no more. Returning to the constant in (4.4) it is of interest to consider its dependence on a. It is best to approach this question from a geometric point of view and in order to do this we need the following observation Lemma 4.1. Let 0 be an open set in R" and for a > 1 associate to it % = { x E R " : Mx,(x) > 1/2a"}.
Then if T,(R"\%) T(R"\O), we have
=
UxeRn\, T,(x),
and similarly for T,(R"\O) =
(i) If (y, t ) E T,(R"\%), then IWy, t)l s 21B(y, t ) n (R"\Q)I. (ii) T,(R"\%) s T(R"\O). Proof. If ( y , t ) E T,(R"\%) there is x g % with ly - X I < at, or x E B ( y , a t ) . Thus IB(x t ) n O I I I W y , t)l s a"lB(y, a t ) n Ol/lB(y, at)l s a"Mx&) s a"/2a" = f, and (i) holds. On the other hand if (y, t ) is in I',(R"\%), (i) implies in particular that there is w E B ( y , t ) n (R"\O) # 0. In this case (y, t ) E T(w),'with w in R"\0, which gives (ii) as well. We are now in a position to show Lemma 4.2. Suppose O is an open set of finite measure and let % be associated to 0 as in Lemma 4.1. Then for a 2 1 and with S , ( F ) = S ( F ) ,
I,"\,
S , ( F ) ( x ) 2dx s 2
I,"\,
S ( F ) ( x ) 2dx-
(4.5)
Proof. From the definition of Lusin function we readily see that
I,.,,
S , ( F ) ( X )dx ~
=-
j
dt m y , t)121B(Y,a t ) n (R"\Wl(at)-" d y t
r,(R"\Q)
(4.6)
and
Now since by Proposition 4.1, T,(R"\%) E T(R"\0) and IB(y, at) n (R"\%)l/v(at)" S f l B ( y , t ) n (R"\O)l/ut" for ( y , t ) in T,(R"\%), the desired conclusion follows by simply comparing (4.6) and (4.7). W We distinguish now two cases, in first place we show Theorem 4.3. Suppose a
3
1 and 0 < p s 2. Then
~ ~ s , (s F ca"(1/P-1/2) ) ~ ~ ~ llS(F)IlP9 where c is an absolute constant.
(4.8)
XII. The Littlewood -Paley Theory
316
Proof. The case p = 2 follows immediately from (4.3). Otherwise let 0,be the open set of finite measure { S ( F )> A} and associate to it %A as in Lemma 4.1. If the reader prefers not to show that 0,is open, the argument given below still works if OA is an open set with measure (arbitrarily) close to { S ( F )> A}. Furthermore, let Sl,= { S , ( F ) > A} and note that
+ Is:,
IsiAl
=I
n’(R”\QA)(
+ J,
(4.9)
say. From Chebychev’s inequality and Lemma 3.2 it follows that J s (s~)-’
J
s,(F)(x)’ dx s 2(sA)-’
J
R”\%
s(F)(x)’ dx. (4.10) R”\OA
Also by the maximal theorem we see that
Thus combining (4.9), (4.10) and (4.11) we get s-”IIS,(F)II;
=
s
J
I G I dAp
I
C09)
CU”
loAldAp+ 2s-’
L0.m)
I
A-’
jRn,OA
S ( F ) ( X dxdh” )~
[OF)
= L + M,
(4.12)
say. Clearly L = ca”llS(F)J1;.On the other hand,
M
= 2ps-’
=
CS-~
jRn
1..
A”-’ dA dx
S(F)(x)’ [S(F)(x),m)
S ( F ) ( X ) ~ + ( ” dx -’)= C S - ’ ~ ~ S ( F ) ~ ~ ~ .
Whence by (4.12) we obtain IIS,(F)JI;s c(a”sP + S”-~)IIS(F)II;and (4.8) follows upon minimizing the right-hand side of the above inequality with respect to s. H As for the case p > 2 we have
(4.13)
where c is an absolute constant.
4. Lusin Area Function and Littlewood-Paley g f Function
317
Proof. Since p / 2 > 1 we may invoke the converse to Holder's inequality
and compare the integrals
" ) norm s 1. First, observe where g is a nonnegative function in L ( p / 2 ) ' ( Rwith tha since
it readily follows that
Thus
1 6 Ils(F)II;II~glJ(P/',~ 6 clls(F)ll;3 IISa(F)'IIp/Z
= IIsa(F)II; c clIS(F)II;,
and we are done. There is yet another important function we consider, namely, the Littlewood-Paley g: function. It is defined forfin Lp(R") and a LittlewoodPaley function q5 by setting F(y, t ) = f * q5r(y) and
Since S ( F ) ( x )4 cg:(F)(x), c independent of F, by the known results for ~ p~ < p ,00. As the Lusin function we have l f&, s cllS(F)IJ,s c ~ ~ g f ( F 1) < for the opposite inequality we have Theorem 4.5. The inequality
XII. The Littlewood - Paley Theory
318
holds, with c = cP,*independent of F, provided either 0 < p < 2 and h > n / p or 2 < p < 00 and A > n / 2 . P m f . We do the case 0 < p < 2 first. Observe that (1
+
(v))-2A 2-'".(&), x - Y l
(4.16)
k=O
where x denotes the characteristic function of the interval [0,1]. Thus multiplying (4.16) through by IF(y, t)12t-" and integrating over R:+' with respect to d y d t / t it readily follows that m
g T ( F ) ( x ) 2s c
2-k(2h--n) S2k(F)(X)2.
(4.17)
k=O
Since p / 2 C 1, (4.17) gives at once m
g f ( F ) ( x ) P< c 12--k(2*--n)p/2 S2k(F)(x)P.
(4.18)
k=O
Whence integrating (4.18) over R" and invoking Theorem 4.3 we readily obtain
II S ( F )IE, k=O
k=O
where the above series converges, since pA - n > 0. On the other hand, if p > 2, then p / 2 > 1, and Minkowski's inequality applied to (4.17) gives
c m
c
Ilg:(F)II; = llg:(F)211p/2
2-k(2A--n) IIs2k(F)211p/2
k=O
-
m
1 2-k(2A-n)IlS2k(F)1; k=O
1 2-k(2*\-n)IlW)II;, m
0 now. These results have numerous applications. We discuss multipliers next. 5. HORMANDER'S MULTIPLIER THEOREM A function m defined in R"\(O) is said to satisfy a Hormander condition of order k provided that lm(5)I
5 s
c
in
R"\(O)
(5.1)
3 19
5. Hormander's Multiplier Theorem and
I
R21 I-," Q
I ~ Q m ( o l ' &=s c
(5.2)
R0
R n / 2 . Then the mulplier operator associated to rn is bounded in L P ( R " ) ,1 < p < 00.
Proof. It is enough to show that rn is an L P ( R " )multiplier for 2 e p < 00. Let TfA(6) = rn( S)f( 0,f E 9 '(R") and observe that on account of Theorem 5.1, S ( G ) ( x )s c g Z ( F ) ( x ) ,and consequently by Thorem 4.5, IIS(G)II, s cllgz(F)IJ,c cllS(F)II, s cllfll,,, provided 2 < p < 00 and k > n / 2 . In other
words, for those functions 1) Tfll, c cllfll, and T admits a bounded extension to LP(R").The case p = 2 follows at once from assumption 5.1.
6. NOTES; FURTHER RESULTS AND PROBLEMS Marcinkiewicz and Zygmund noted in 1939 that for an arbitrary linear operator T which is bounded in L P ( R " ) with , norm 11 Tll. the inequality
II(cI~f2)1'zllP I1 7-11ll~~l~12~1'211p
(6.1)
Xzz. The Littlewood - Paley Theory
322
also holds. This chapter deals with variants and extensions of this estimate. Rubio de Francia [1982] recently observed that some results of Maurey concerning factorization of operators can be used to show that vector-valued inequalities are, to some extent, equivalent to weighted inequalities. In particular, he showed that given a sequence {'I;.} of sublinear operators bounded from L P ( R " )into L q ( R " ) ,and given a = p / r , p = q / r < 1, the ' ) ~ if' ~ and ~ [ only ~ if for every estimate ~ ~ ( ~ ~ T J J rC) Cl '~r ~~ (~ ~q ~ ~ Jholds nonnegative function u in Lp'(R")there exists a nonnegative function U in L*'(R"), l / a + l / a f = l / p + l/pf = 1, such that (1 tYlla,C [lulls. and jRnITf(x)l'u(x)dx s C ! ~ ~ I ~ $ ( X ) I ~dx, ~ Yall ( Xj ). A similar statement holds in case a,p < 1. In the general context of Banach space valued CZ singular integral operators, the theory was developed by Benedek, Calderh, and Panzone [ 19621 and Riviire [ 19711. The proof of Theorem 5.1 is due to Stein. Further Results and problems 6.1 Prove (6.1). (Hint: It suffices to prove the estimate when6 = 0, j 3 N, some large N. Let C denote the unit sphere in R N , put f ( x ) = ( f i ( x ) ,. . . , ~ N ( x ) ) , T f ( x ) = (Tfl(x), . . . , T f N ( x ) )and observe that by the linearity of T we have T(y' . f ( x ) )= y f . T f ( x ) .Thus
We now invoke the following property: if w E RN\(0), then . w J pdy' = cII wll? with c # 0 independent of w ; the possible dependence on N and p is irrelevant here since c cancels itself out. Thus integrating the above inequality over C it follows that r r 6.2
Let {Ik} be a sequence of disjoint, open cubes in R" and put& A simple computation gives that
=
x,,.
where yk denotes the center of z k and T~ is a simple modification of the classical Marcinkiewicz integral of order r corresponding to the cubes {Ik}. More precisely, if dk = diameter of Ik and 7 2 n ( r - l), then
c
d ;+" d;+"
IX - ykln+"-k
6. Notes; Further Results and Problems is bounded by case gives
T,(x)
323
(cf. 5.26 in Chapter V). Theorem 1.1 applied to this
(i) F o r l / r < q < W , 1 1 T r l l z S CClZ/cl, r A}l 5 cCIZkl, and (ii) A 1 / r l { ~> (iii) If I is a finite cube with I k s I , then over I (extrapolation from (i)).
u
T~ is
exponentially integrable
This interesting application of Theorem 1.1 is also due to Fefferman and Stein [1971]. 6.3 There is, of course, a weighted version of Theorem 1.1. More precisely, with the notation of that theorem and d p ( x ) = w ( x ) dx we have (i) If 1 s p < 00, there is a constant c such that APp({IIMfl),r> A}) G if and only if w E Ap (ii) If 1 < p < 00, thereis aconstant c such that 11 ~ ~ M f ~S ~cII l~~ ~~ ~ fL ;[ if and only if w E Ap. (iii) If w E A , and I is a finite cube then IIMfll;r is exponentially integrable over I (with respect to dp) whenever Ilf(x)ll,r < 00 and supported on Z.
cJI
llflllr114
These results are included in the work of Anderson and John [1980]; Heinig [1976] considered results in the range 0 < p < 1 for w E Al . 6.4 Anderson and John observed that (i) and (ii) in 6.2 hold with the Lebesgue measure replaced by dp ( x ) = w ( x ) dx provided w E A,, and also (iii) holds provided that w E A,. 6.5 An operator T defined in some Lp space is said to be linearizable if given any f in Lp there is a linear operator U = U, on Lp and that I Tfl = I Ufl and I Ugl S I Tgl for every g in Lp. Maximal operators corresponding to a sequence of linear operators and operators of the form T f ( x ) = ( ~ , ~ T , , , f ( ~ ) ~ ' 1d c w )r ~ 1, the dyadic decomposition of R ” is obtained by taking the product of the dyadic decomposition of R in each direction. In other words, we write R ” as the union of disjoint “rectangles,” each of which is a product of intervals which occur in the dyadic decomposition of the coordinate axes. This family of rectangles p is denoted by A. Show that if S, denotes the operator corresponding to the mutiplier xp and f E L p (R “ ) , 1 < p < 00, then II(~,,AlSpf12)1/211p Ilfllp. (Hint: As usual it suffices to prove
-
-
325
6. Notes; Further Results and Problems
ll(C,,als$12)1/211p cllfllp, 1 < p < 00, and then dualize. There are several proofs of this. A way to go about it is by using induction over the number of variables and Theorem 2.5 in its full strength. Let Zl, Z2,. . .be an arbitrary enumeration of the dyadic intervals of R and observe that each p in A is of the form Zm, x * * x 1,. ; we call such a rect_anglep m , rn ( ml,. .. ,rn,,). Now as in 6.8 we observe that (S,J (5) = ~ ~ , ( 5 ) 4 ( 5 ~ / 2 "* ' ~ + ~ ) 3(5n/2mn+1)j.(5). Thus IISAfllP d IIG(f)ll, where G ( f ) ( x )= (~m+,,l,...,mn)lf * $m(x)12)1/2 and $ r n ( X ) = 2 " 1 4 ( 2 " 1 ~* ~- ) 2"n4(2"nx,,).Let H 1 = 12(2"-'), the space of ( n - 1)-tuples of square summable sequences, m' = (m2,. . . , rn,,), and apply 6.8, i.e., the one-dimensional result, to f * $,,,(x) as a sequence of functions of x1 indexed by ml and with values in Hl. This idea is from Rivibre's work [1971].) 6.10 The following limiting case of Theorem 4.5 holds: suppose 0 < p < 2, and A = n / p , then t P J { g f ( F> ) t}l d cllS(F)IIi. (Hint: let E = { M ( S ( F ) ' ) > t P / 2 } , = { S ( F )> 2"k'pt} and %k = (kf(X0,)> 1/2"k+1}. First notice that %k c E, all k. Indeed, if x E %k, then there is a ball B(y, r ) which contains x and such that 1B(y, r ) n 4 1> IB(y, r)l/2"k+'.Therefore =A
-
-
I
S( F ) (x ) dx ~ 2 tp'2
S ( F ) ( X ) ~3~ X B(w)
and
x
E
E.
B(w)nQk
On the other hand, by Chebychev's inequality we also have that I = t 2 ) { x E R"\E: gf(F)(x) > t}l =Z m
r
m
2-k(h2--n)
d c k=O
S ( F ) ( x ) 2dx
R"\ffk
Let h be the smallest integer k so that x E R"\Ok. Then the sum in the above integral is of order 2-h'App-n) and from the definition of h it also follows ~ -= "''" that S ( F ) ( x )d 2"h'pt. Thus I s cl,. S ( F ) ( X ) ~ ( S ( F ) ( X ) / ~ ) - ~ ' "dx Ct~("2-n)/n S ( F ) ( X dx. ) ~ These bounds combine to give the desired conclusion. The proof given here is due to Aguilera and Segovia [1977] and extends some results of Fefferman [1970].) 6.11 Given f E Y ' ( R " )and a Littlewood-Paley function 4, let F(x, t) = f* and Put
I,.
326
XII. The Littlewood - Paley Theory
where 4 is integrable. Determine the continuity properties of G ( F ) as we did for the particular case of the function g f ( F ) . The results of Madych [1974] are relevant here. 6.12 Let m be a measurable function which verifies ~ ~ m B~ and ~ m s J,ldm(5)1 C B for every dyadic interval I of R Then rn is a bounded depends only on B and multiplier in L P ( R ) ,1 < p < 00, with norm w!cih p . (Hint: For f in C F ( R ) put $([) = m ( f ) f ( ( ) and for an interval p in the dyadic decomposition A of R _and w E p setpp,w(y)= x ( { y E R : y E p, y < w } ) . Furthermore, let (SP,,f) (5)= ~ , , ~ ( & ) f ( fand ) if p = (2k,2k+1), say, set
Since F'( w ) = f(w)eixwa.e. integrating by parts the expression
and
To complete the proof it now only remains to bound the integral above, and this will follow from the estimate
with a-constant c indetendent of N. In first place notice that for 5 E p, ( S J ) ( y ) = (Sp,c(Spfl)(y), so that actually SP,*f is a partial sum of SJ
6. Notes; Further Results and Problems
327
NOW,divide each pi whch appears in the sums of (6.3) into k equal parts by partitions [ ; , j = 0,1,. . . , k, i = 1,. . . , N. By 6.8,
which letting k + co gives (6.3) and we are done. For further details and related results the reader may consult Kurtz's work [ 19801). 6.13 We think, now of R" as divided into 2" "quadrants" by the coordinate axes, the first quadrant being the set {x = (x,, . . . ,x,) :xi > 0, i = 1,. . . ,n } and so on. Let m be a measurable function which verifies 11 m ) ) m S Bym E C" in each quadrant of R" and so that
. . ,5"). If 1 < p < 00, show that rn is a bounded multiplier in L P ( R " ) with , norm which depends only on B and p . (Hint: The proof is similar to that of 6.12. Indeed we decompose Spg into a sum of 2" pieces now, each of which is handled as before. This result, which has many important applications, is known as the Marcinkiewicz multiplier theorem.) 6.14 Rivikre [ 19711 observed that there is a theory of vector valued multipliers as well. These general results have many interesting applications. p any dyadic rectangle in Rk, and any permutation of (if1,.
CHAPTER
XI11 The Good A Principle
1. GOOD A INEQUALITIES A good A inequality is a principle which allows us to derive norm and even local or pointwise estimates of one operator in terms of another provided they satisfy an a priori relation of probabilistic or measure theoretic nature. We have already considered some instances of this principle in 4.20 in Chapter VI and Theorem 4.4 in Chapter X.
Definition 1.1. Given a positive, doubling, regular Bore1 measure p on R" we say that the operators TI, T2verify a good A inequality with respect to p provided the following three properties hold: (i)
TI, T2 are sublinear and positive.
(ii) { Tlf > t } is an open set of finite Lebesgue measure for each f in C F ( R " )and t > 0. (iii) If a ball B contains a point x where T l f ( x )s A, then to each 0 < 7 < 1 there corresponds y = y ( TI, T 2 ,7 )independent of A, B and f
so that P({Y
E
'
B : TIf(Y) 3 4 T2f(Y) s
rA))s T P ( B ) .
(1.1)
(1.1) expresses the control in measure alluded to above and the norm estimate is given by Theorem 1.2. Suppose TI, T2 verify a good A inequality with respect to p and assume that 11 TlfllL:< 00, 0 < p < CO, for f in C:(R"). Then there is a constant c = c,,~ independent off E C:(R") so (hat
II Tlfll L:
c 328
1I T*fllL.:
*
(1.2)
1. Good A Inequalities
329
Prmf. Let f E C,"(R")and assume that 11 T2fllL;< 00, for otherwise there is nothing to prove. OA = { Tlf> A} is an open set of finite Lebesgue measure and consequently to each y in OA we may assign a ball B(y, r,,) centered at y and of radius r,, so that B(y,
s OA,
B(y, 3ry)
(R"\6A)
f
0.
(1.3)
Let = {Tif> 3A,
%A
T2f s YA},
( 1.4)
where y is a constant yet to be chosen; clearly, %A 5: 0,.To estimate ~ ( 9 . l ~ ) we consider K, an arbitrary compact subset of a,,. Since K c_ B ( y , r,,) there exist finitely many balls among the B ( y , r,,)'s so that actually
uYEoA
K
s Ufinitely many B(Y, ry).
(1.5)
Let now B, ,. . . ,Bmbe a disjoint subfamily of the family in (1.5) such that m
K E
u 3Bj,
j=1
and let 0 < 7 < 1 be another constant yet to be selected. Since we will invoke the good A principle, y in (1.4) will automatically be determined by property (iii) in Definition 1.1 once 7 is chosen. Now, since K E %, we can sharpen (1.6) to
K E
ij( { y
j=l
E
3 ~ j ~: 1 f ( y>) 3 4
s
~ 2 f ( y )
y~}).
(1.7)
Since it is clear from (1.3) that each ball 3Bj contains a point xi where T , f ( x j )s A, by the good A principle (still with 7 to be chosen and with y to be determined by (1.1)) on account of ( l . l ) , we have that
where the last estimate follows since p is doubling, and c and 17 are independent of A. Furthermore, since the Bj's are disjoint balls totally contained in OA, and K is an arbitrary compact subset of %A, from (1.8) we immediately see that
330
XIII. The Good A Principle
This is all we need to complete the proof. Indeed, first observe that by (1.9)
= C + P I I T1.mE+ ( 3 / y ) p 1T 1 ~~IIQ.
(1.10)
We now choose 77 so that ~ 7 7 = 3 ~1 and y so that (1.1) holds. With this choice, and since by assumption IITlfllL;< 00, we may rewrite (1.10) as 11 Tlfll%;S 2 ( 3 / y ) p11 Tzfll p,L; and we have finished.
2. WEIGHTED NORM INEQUALITIES FOR MAXIMAL CZ SINGULAR INTEGRAL OPERATORS
In considering the estimates of interest to us, it is important to decide what restrictions to impose on the weights. In view of the following observation our basic assumption will be the Ap condition. Proposition 2.1. Let Hf denote the Hilbert transform off: If p is a nonnegative, regular, Bore1 measure and H is bounded from LP,(R) into wk-LP,(R), 1 s p < 00, then (i) p is doubling, G ~ ( M , ( l f l ~ ) ( x ) )and "~, (ii) Mf(x) (iii) p is absolutely continuous with respect to Lebesgue measure and dp(x) = w(x) dx, where w is an Ap weight. Proof. Fix an interval I and let I, and Il denote the abutting intervals to I, of equal diameter, which lie to the right and to the left of I respectively. Clearly (i) follows from the estimates
p ( I r ) ,~ ( I I )C P ( I )
(2.1)
2. Maximal CZ Singular Integral Operators
331
with c independent of I, which we now prove. For a nonidentically 0 function f in LP,(R) consider the restriction of Ifl to I. Clearly lAxl E LP,(R) and for x E I, we have H ( l f l x r ) ( X ) 3 ((1/211l) Jrlf(u)l dY)xr,(x). I r E {IH(IAxr)l> (1/2111) Jrlf(Y)l d ~and )
Obviously a similar estimate holds for p ( l l ) .By puttingf = in (2.2) we immediately see that p ( I , ) S cp(1) and (2.1) holds. Now that we know p is doubling we may rewrite (2.2) as
and (ii) follows. Moreover, since p is doubling and regular, by virtue of (ii) and Theorem 1.1 in Chapter IX the Hardy-Littlewood maximal operator maps LP,(R) into wk-LP,(R) and, by Theorem 2.1 in Chapter IX, p is absolutely continuous with respect to Lebesgue measure and dp ( x ) = w ( x ) dx, w E A p . We pass now to consider the weighted norm inequalities for CZ singular integral operators. Since we will invoke the good A principle it is convenient to deal with the maximal CZ operators directly. Theorem 2.2. Suppose the CZ kernel k verifies assumptions (i)-(iv) in Chapter XI and in addition
(v) Ik(x)(s c5(x(-",x f 0. (2.3) Furthermore, let T* denote the maximal CZ singular integral operator associated to k and let d p ( x ) = w ( x ) dx, w E Ap.Then there is a constant c = cp such that
II T*fllL:: cllflL: ,
1 < P < 00. (2.4) Proof. We verify that the good A principle holds for TI = T* and T2 = M; in other words, we check that properties (i)-(iii) in Definition 1.1 hold for these operators. (i) Suppose f E C,"(R")and observe that since T* is of weak-type (1, l), = I{ T*f > A}( is finite for each A > 0. To see that OA is open let then 10A( {x,} be a sequence of points in R" so that T*f(x,) =sA and x,,, + x ; we claim that also T * f ( x ) 6 A. First, put TxX)
=
I
k ( y ) ( f ( x - y ) - f ( X m - Y ) ) dy
lYl=-E
+
I,,.
k(Y)f(xm - Y ) d~ = I r n + J m ,
XIII. The Good A Principle
332
say, and note that I, + 0 as rn + 0O and IJ,,,] c A, all rn. The latter is easy to check since lJ,l = ITEf(xm))ld A , all rn. As for the former observe that for each E > 0 the function k E ( y )f( ( x - y ) - f ( x m - y ) ) tends to 0 as rn + my is integrable, and is pointwise majorized by the integrable function C ~ ~ , ( Y ) I X ~ (N ~ ,=~ N ) ((Yf )sufficiently , large. Thus by the Lebesgue dominated convergence theorem it follows that I,,,+ 0 as rn + 00. This in turn implies that I T f ( x ) (d A for every E > 0 and consequently T * f ( x )d A as well. (ii) Let B = B(0, R) be a ball with large enough radius so that it contains the support o f f E C?(R") in its interior. Then
say. To estimate I note that by Holder's inequality and Theorem 6.2 in Chapter XI,
Since w E RH,, for some r > 1, with this choice of r it follows that I < co. As for J, observe that for X E R"\lOB and by (v) above T * f ( x ) d jlylrRlk(x- y)ll f ( y ) l dy d c5IxI-"Ilfll1.Consequently, by Proposition 4.4 in p 00. Chapter IX J d c ~ R ~ \ l o B ( dx p~(-x~) c (iii) This is the heart of the matter. Suppose B contains a point w so that T * f ( w )S A and let E = { x E B: T * f ( x ) > 3A, M f ( x ) d yA}, where y is a constant yet to be chosen. Recall that by Proposition 4.4 in Chapter IX there are constants c = cp, r, independent of E, B, so that p ( E ) / p ( B )d c(IEI/\BI)~.Thus it suffices to verify (iii) for the Lebesgue measure instead. Write f = fxzB + fXRn\2B = f l + f 2 , say and observe that E E { x E B: T * f l ( x )> A } u { y E B : T*f2(x)> 2A, M f ( x ) S yA} = El u E 2 , say. We will show that, given 0 < r] < 1, there is y = y ( 7 ) sufficiently small that simultaneously lEll d r](BIand E2 = 0;once this is achieved the proof will be complete. In the first place observe that since T* is of weak-type (1,l) AIEll
Cllfilll =
J
2B
lf(v)l dY.
(2.5)
-
Let I be a cube concentric with B such that 2B c I and ) I [ IBI. Then the right-hand side of (2.5) is dominated by
#
[ ) f ( y ) I dy d clBl iFf Mf d cIBI inf B MJ:
(2.6)
2. Maximal CZ Singular Integral Operators
333
If E = 0 there is nothing to prove. If on the other hand E # 0, then infs Mf d infE Mf d yA. Consequently, by (2.5) and (2.6),
IElI 4 B I , (2.7) where c is an absolute constant independent of E, B, and f: Next we show that E2 is empty provided y is judiciously chosen. For this purpose we estimate TAX) =
+ +
(I, (I,
k ( x - Y ) f 2 ( Y ) dY X-Yl>E
I, Lw-yl>E w --y
k ( x - Y l f d Y ) dv -
W-yJ>E
I,w-y,>c
k ( w - Y ) f i ( Y ) dY
=
>L
k ( x - Y ) f z ( Y ) dY)
k ( w - Y ) f 2 ( Y ) dY)
I +J + K
say. We consider J first and observe that
I,
IJI
Ik(x - Y ) - k ( w - y)I Ih(y)l dy
W-yl>
E
where 3 E E, and in particular Mf(3)d yA. Now Ix - wI of By and from (2.8) we see at once that
S
cry r = radius
where cJ is independent of B andf: To bound K we distinguish two cases, namely, (i) E < 2r and (ii) E 2 2r. If (i) holds, then fi = f in the integral and IKI d T * f ( w )d A. If on the other hand we are in case (ii), then we may also assume that E 3 r for otherwise the part of the integral between E and r vanishes. Thus
11,
IKI
k ( w - Y ) f 2 ( Y ) dY -
w-y)>r
+
,!I
I,
w-yl>
k ( w - Y l f A Y ) dYl 3r
k ( w - V ) f i ( Y ) dyl = lKll+ IK2L w-yl>3r
say. Clearly lKzl d A. As for lKll it is bounded by
I
r 0. It is at this point that we pick y so that y ( c I + c, + c K ) < 1, in addition y must be small enough so that cy < 7 , where 0 < 7 < 1 is given and c is the constant in (2.7). In other < 2A, which in turn words, we have proved that for x in E,, supE(Tef2(x)( implies E2 = 0, and that lEll d 71BI.
+ +
+
3. WEIGHTED WEAK-TYPE ( 1 , l ) FOR CZ SINGULAR INTEGRAL OPERATORS
Since the proof follows along the lines of Theorem 2.14 in Chapter V, we shall be brief.
Theorem 3.1. Suppose the CZ kernel k verifies the assumptions of Theorem 2.2 and d p ( x ) = w ( x ) dx, w E A , . Iff E L,(R"), then T*fE wk-L,(R"), and there is a constant c independent off such that A P ( { T * f > A})
S cllfllL,,
A > 0-
(3.1)
Proof. Let ( 4 ) be the Calder6n-Zygmund (interval) decomposition of Ifl at level Ah, where A is yet to be chosen, and set R = U 4, and f = g + b, where g and b are the good and bad parts off corresponding to the 4's.
3. Weighted Weak-Typefor CZ Singular Integral Operators
335
respectively. Clearly T * f ( x ) c T* g ( x )+ T * b ( x ) and it suffices to show ( 3 . 1 ) with f replaced by g and b on the left-hand side there. The estimate for g is straightforward. Observe that since w E A l ,
Therefore, since also w
AZtL({T*g> A))
E
AZ,by Theorem 2.2 and (3.2) it follows that
cllglli;
s cAA llfIlL.+, which leads to an estimate of the right order. (3.2) also implies that if = u j ( 2 4 ) , then p(a1) s cII f l l L , / A , and consequently it is enough to show that AP.({R"\Rl: T*b For this purpose fix x in R"\O,,
cJ
E
'A)) c
CllfllL,.
(3.3)
> 0, and consider
c
T N ~ =)
j
k ( x - Y M y ) dy.
(3.4)
(R"\W%jc))nI,
We separate the 4 ' s into three subfamilies, to wit (i) those 4's contained in R"\B(x, E ) , (ii) those 4's contained in B(x, E ) , and (iii) the rest. Since b ( y ) dy = 0 and Ib(y)l s If ( y ) l + cAA, the sum in (3.4) extended over the j ' s in the first family is readily seen to be dominated, in absolute value, by
I,
where yj denotes the center of 4. All summands corresponding to the second family vanish. As for the third family, note that if l, belongs there, then 4 n B ( x , E ) # 0 and 4 n (R"\B(x, E ) ) # 0 simultaneously. Moreover, since E > 2Lj = 2 (sidelength of I j ) , there is R > 0 so that I j s B(x, 2R)\B(x, R / 2 ) . Now let
u
XIII. The Good A Principle
336
and observe that (3.7)
From (3.6) it also readily follows that lcil s cAA.
(3.8)
To estimate the sum in (3.4) extended over the j ' s in the third family we note that it is bounded, in absolute value, by
x I k ( x - U ) - k ( x - yj>I(If(Y)I + CAA)d~ IJ
+ cAA
I
Ik(x - Y)I dY-
(3.9)
B(%2R)\B(x,R/2)
Furthermore, observe that the last integral in (3.9) is dominated by c5 ~e(x,2R)\B(x,R/2)1~ - y ( - " dy s c, and consequently the last summand there by cAA. (3.10)
Thus choosing A so that CA< f, where c is the constant in (3.10), and combining (3.5) and (3.9) it readily follows that T * b ( x )S
xI
~ k (x y ) - k ( x - Yj>l(lfl+
i
A
dy + 2
= 44x1 + A/2,
say, and consequently {T*b > A} c { + ( x ) > A/2}, x by Chebychev's inequality we get that
E
R"\n,. Moreover,
(3.11)
say. We take a closer look at the inner integrals in (3.11). If Lj = sidelength of 4, then Ik(x - y) - k ( x - y j ) (< c 4 ( L j / ( x- yl"+l),and we claim that each
4. Notes; Further Results and Problems
337
integral is bounded by
Indeed, first observe that
c 02
6
(2kLj)++1)P (B (Y, 2kLj))*
k=l
Also, since w E A l , by (2.9) in Chapter IX, p ( B ( y , 2kLj))/(2kLj)n s c p ( B ( y ,L j ) ) / L ; < cp(4)/141, and consequently N c cp(4)/141Lj, as anticipated. Substituting now (3.12) into (3.11), gives that
s cllfllL: + C A P 1 s cllflL:,
and the proof is complete. 4. NOTES; FURTHER RESULTS AND PROBLEMS
In 1970 Burkholder and Gundy introduced a technique that has since been used quite effectively in establishing the continuity of various kinds of operators, namely the so-called good-A inequalities. The main application of this idea in this chapter, following Coifman and Fefferman [19741, is to the boundedness of CZ singular integral operators in weighted Lp spaces. In the particular case of the Hilbert transform this result is due to Hunt, Muckenhoupt and Wheeden [ 19731. Further Results and Problems Suppose j - C~? ( R " ) and d p ( x ) = w ( x ) dx, with w E A,. Show that IIMfllq < cllMZZfll~:,1 < p < 00, where c = cP+ is independent of f: ( H i n t : It suffices to show that Tl = M and T, = M # verify conditions (iii) in Definition 1.1, and this is not hard. Since w E A,, as in Theorem 2.2, we see that in fact we may restrict ourselves to the Lebesgue measure.
4.1
XIII. The Good A Principle
338
We show then that given an open ball B and y > O , [El = I{x E B : M f ( x ) > 3A, M # f ( x ) G yA}l s cylB(, where c is a dimensional constant independent of B, A, y, f, provided B contains a point w where M f ( w ) s A. Note that this last condition implies that S A for every open cube I containing B, so if now I denotes a fixed open cube concentric with and containing B, 1I1 IB), and x E E, then also M ( f X I ) ( x )> 3A and M ( ( f - , f r ) X , ) ( x ) > 2A. Therefore either E is empty or else (El s I { M ( ( f - h ) X r ) > 2 ~ 1 1s cIrlf(y) -frI ~ Y / A = c y l ~ l - Since for f ( x ) = c a.e. we have M f ( x ) = c and M # f ( x ) = 0, it is clear that the result does not hold in general unless some restrictions are placed on $ Some of these restrictions are: f~ L z ( R " ) ,some q < p, Fefferman-Stein [1972], I{Mf> &}I < 00 for each E > 0, Calder6n and Scott [1978], and inf(1, Mf)E Lp, JournC [1983].) 4.2 The assumption )ITlfllL;< co is essential for the validity of Theorem 1.2. The following pertinent example is due to Miyachi and Yabuta [1984]: Suppose a > 1, 6, p , a > 0 and 1 S / p s a'. Then there exist nonnegative, measurable functionsf, g on R" such that I { f > ah, g s yA}lG y ' I { f > A}l for all y, A > 0, llgllp < 00, A}l = O ( e " - " ) as A + 0, and Ilfll, = CO. To see this choose a disjoint sequence { E k } of measurable subsets of R" with lEkl = eak" and define { y k } by y",EkI = IEk-ll. Then put f = a - k and g = a-(k+l)yk+l on Ek and f = g = o outside Uz=,,Ek. 4.3 Under the usual assumptions on the vector valued CZ kernel k, Theorem 2.2 and 3.1 remain valid. In other words, the Littlewood-Paley and Lusin functions are bounded in LP,(R"),d p ( x ) = w ( x ) dx, w E Ap, 1 < p < 00, and of weak-type ( 1 , l ) on LP,(R")when w E A l . As for the g: function, one must be careful with range of the parameter A. These and related results are discussed in full detail in the work of Stromberg and Torchinsky [1980]. An immediate corollary to these results and Theorem 5.1 in Chapter XI1 is a weighted version of the Hormander multiplier theorem. Another application is to the Marcinkiewicz multiplier theorem 6.13 in Chapter XII. For instance, the proof given in 6.12, Chapter XII, for the case n = 1, extends immediately to the weighted setting, provided w E Ap,by simply using the corresponding weighted inequalities in estimates such as (6.2) there (cf. Kurtz [1980]). 4.4 The conclusion of 6.5 in Chapter XI1 also applies to the Lusin and g: functions, for instance. We must be careful, however, with the range of values of A (cf. Rubio de Francia [1980]). 4.5 The following estimates hold for the Calderbn-Zygmund operators T introduced in 8.11-8.12 in Chapter XI: Ap({ITfl > A}) G cllfllL,, d p ( x ) = =)4 x 1 dx, w E A p , 1 < P < w ( x ) dx, w E A,, and IITfllL; cllfllL:, 4 . 4 ~ 00. Also, and with the same assumptions on T and p as above, Ap ({ 11 Tfk11 I' > A}) cII IlfkllI'IIL, and 11 IITfkIII'IIL; cII IlfklIl'IlL;, < < O0 (cf*
-
CIBIYA/ A
+
I{f>
4. Notes; Further Results and Problems
339
Hernandez [1984]). As for the weighted version of Theorem 1.3 in Chapter XII, cf. Andersen and John [ 19801 and Jawerth and Torchinsky [1985]. 4.6 It is convenient to have a way to decompose open sets in R" so that it will then be easy to set the stage for the good A procedure to work. A way to go about it is as follows: Let R be an open set of finite Lebesgue measure in R". Then there exists a collection of closed cubes {Qk} so that (i) UkQk = R, (ii) if I k = interior of Qk, then 4 n I k = 0 , j # k, and (iii) there are absolute constants cl, c,, independent of R, such that c1 diam Qk S dist(Qk, R"\R) s c2 diam Qk. This is the proof First divide R" into a mesh of congruent closed cubes, with pairwise disjoint interiors, each of measure 3lRl/2; in this way none of the cubes may be totally contained in SZ. Next discard any cube in the mesh which does not intersect R and subdivide each of the remaining cubes into 2" congruent closed cubes, again with disjoint interiors, by bisecting the sides. Once again discard any cube in this new mesh which does not intersect R and separate those cubes Q of this mesh which are totally contained in Q and which in addition verify . . . ,etc. As for those dist(Q, R"\R)/diam Q 2 1, and rename them Q1, Qs, cubes which are left, and these are cubes which either intersect both R and R"\R, or they are totally contained in R and dist(Q, R"\R)/diam Q < 1, subdivide them again into 2" congruent cubes as before and repeat the selection process. It is not hard to check that (i)-(iii) above hold for the family {Qk} thus selected. It is also possible to be a bit more careful and give a proof that works for arbitrary open sets R c R", whether they have finite measure or not. This result is called the Whitney decomposition and has numerous applications.
CHAPTER
XIV Hardy Spaces of Several Real Variables
1. ATOMIC DECOMPOSITION When discussing in Chapter XI1 the norm relations between the S and < 00. This means, in particular, that the Hormander multiplier theorem remains true for those functions whose Lusin integral is in L P ( R " )0, < p < 00. When 1 < p < 03 these classes of functions coincide with the usual L P ( R " )spaces, but what can we say in case 0 < p d 1? More precisely, which classes of functions, or even tempered distributions, are characterized by the fact that their Lusin integral is in L P ( R " ) ,0 < p s l ? They are precisely the Hardy H P ( R " ) spaces of several real variables. In fact, elements in these spaces have the remarkable property that they can be written as sums of elementary components, or atoms. In a sense a very sophisticated Calderbn-Zygmund decomposition holds in these spaces. We begin our discussion by giving examples of atoms and then proceed to show how they combine to span the Hardy spaces. g: functions, we presented our results in the range 0 < p
Definition 1.1. Let p, q, N be subject to the following conditions
where [ ] denotes, as usual, the "greatest integer not exceeding" function. We then say that a function a is a (p, q, N) atom provided that (i) a ( x ) = 0 off I (some open cube depending on a), (ii) Ilallq < 00, and 340
1. Atomic Decomposition
341
(iii) I , x'a(x) dx = 0, for all multi-indices rw with la1 d N. To somehow normalize the atoms, we introduce the quantity la/,or atomic norm of a, by la1 = infl4 I
I/p--l/q
Ilallq,
(1.2)
where the infimum is taken over those cubes I which verify (i), (iii) above. We then have
+
Proposition 1.2. Let a be a (p, q, N ) atom and suppose that is a Littlewood-Paley function which satisfies, in addition to conditions (i)-(iii) in ) Definition 3.1 in Chapter XII, the following property: E C N + ' ( R n and ID"+(X)ls c(1 + IxI)-("+e+N+')for all multi-indices rw with la1 = N + 1, and some E > 0. Then, if A(x, t ) = a * & ( x ) , S ( A ) E LP(R"),and there is a constant c = cP,$ independent of a so that
+
IIS(A>IIPs cl.1.
(1.3)
Proof. By translating a if necessary we may assume that the cube I, which verifies (i)-(iii) above, is centered at the origin. We estimate
say. To bound J, note that, by Holder's inequality with indices q/p and its conjugate and by (4.4) in Chapter XII, J s ~ 2 1 / ' - p ~ q ~s ~S(A)~~~
(1.4)
which is an estimate of the right order. To bound K requires some work. By Taylor's expansion formula we have that $(x
Wol
+ w, =
+ R N ( X W)Y ~
-D"+(x)
(1.5)
lalSN
with IRN(x,w)l s c1wIN+' suplD"+(x + vw)I, where the sup is taken over IrwI = N + 1 and 0 d 7 s 1. Therefore, in our case
Now, by (iii) above and (1.5) it is clear that
-Ic La !( y ) m D " + ( $ ) d ~
A(y,t)=[,a(w)t-"(+(~) =
J-,a(w)t-"R,(;,-) Y --w
o1IS
dw.
N
(1.7)
XIK Hardy Spaces of Several Real Variables
342
We must estimate the integrand in the last integral in (1.7)for w E I and (y, t ) E T(x), x 21. Observe first that under these conditions (t+Ixl) 0, + 1, then F ( x , t) = f * c$,(x) is well defined and limt+mlF(x,t)l = 0, uniformly in x. (ii) If II, is a Littlewood-Paley function and in addition ID"$(x)l s c ( l + IxI)-(n+E+14+l) some E > 0, la1 s N + 1, and G(x, t) = f * II,,(x), A k ( x , t ) = ak * II,*(x), then S ( G )E L p ( R " ) and there is a constant c, independent o f f , such that [IS(G)IIp cA. la1 s N
Proof. The unconditional convergence of the sum C ak to a distribution f follows at once from the fact that if c$ is as in (i) above, then
I
= xlak(c$)l
2k}, k = 0, & I , . .. and 4 = { M ( X E > ~ )f}. Note that by the Lebesgue differentiation theorem Ek E o k a.e. and by the maximal 41s clEkl, c independent of k. Observe that Ek decreases from theorem 1 R to 0 as k increases from -00 to 00, and similarly for 0,. For each (y, t) in the upper half plan R: let I ( y , t) = { w E R : ly - wI < t} denote the interval centered at y of diameter 2 t and let 4: R: +. 2 be the function given by + ( y , t) = largest integer k so that
'
(1.21) II(y, t ) n E k l Il(y, t)1/2Clearly, by the above observations 4 is well defined. We list now some properties of the sets + - ' ( k ) , to wit:
(a) The sets 4-'(k) are pairwise disjoint and U k 4 - ' ( k ) = R:. (b) If ( y , t) E 4 - ' ( k ) , then I ( y , t ) E o k . (C) If ( y , t ) E + - ' ( k ) , then Il(Y, t ) n (R\Ek+l)l 3 Il(Y, t)1/2(a) is obvious. Also, if (y, t ) E 4 - ' ( k ) , then (1.21) holds and consequently inf1(,,,)MxEk> 4, thus proving (b). (c) is equivalent to ( I ( y ,t) n Ek+'(S l l ( y , t ) ( / 2 , which also holds on account of the definition of 4. Since is open it may be written as a countable, disjoint union of open intervals, o k = U j 4,kr say. This allows to localize the situation at hand by setting 'T;;k = {(y, t ) E + - ' ( k ) : y E 4 . k ) . It is clear that the q , k ' s are also pairwise disjoint and that u j q,k
= d-'(k),U j , k
q , k = R:.
+
These observations are all that is needed to complete the proof. Let be the function constructed in Proposition 1.7 corresponding to the value N in the hypothesis and put
XIK Hardy Spaces of Several Real Variables
348
We claim that the aj,k’s satisfy the following three properties, to wit: (i) If aj,k(x) # 0, then x E 3 4 , k . (ii) N moments of a,, vanish. (iii) [laj,kllqs C2k14,k11’q,1 < q < 00, c independent ofj, k. Assume for the moment that (i)-(iii) have been verified. Then, since 14,i,k11’P-1’qllaj,kIlq s C2k14,,k11’p,we immediately see that
d
Whence ( 1.22) j,k
k
To bound the sum in the right-hand side of (1.22) we note that it equals
I
c
jsNl,(kl=zN2
P
;I,,,,.,
dx
2kp
I S(G-
=
2”dx s c
R (k:S(x)sZk}
-0
as
IRn
S(x)pdx.
N1,N2+w.
Thus we will be finished once we show that f = g; this is not hard. Indeed, by the unconditional convergence of 1 aj,, we readily see that its sum actually is I,: v(y, t)t,bt(y- x)/t dydt(Y’). Therefore, by Proposition 1.9 it follows that f(7) = g ( v ) for every Schwartz function 7 with vanishing integral. By Proposition 1.6 in Chapter X this means that f - g is a polynomial 9 (that is, the Fourier transform of a distribution supported at the origin). But by assumption and Proposition 1.4 both the Poisson integrals off and g go to 0 as t + co. This implies that for every x, 9 * P r ( x )+ 0 as t + 00, and so 9 0. Thus to complete the proof it only remains to verify that properties (i)-(iii) of the aj,k’shold. (i) If aj,kz 0,there exists (y, t ) in T , k , and y in 4 , k , so that ly - XI s t for otherwise + ( ( y - x)/ t ) vanishes identically. Moreover, since I ( y , t ) E o k in this case, we actually have that I ( y , t ) G 4,,k and consequently t s (diam 4,,)/2. Thus letting yj,k = center of 4&, we see that Ix - Yj,kl s Ix - yl and we have finished.
+ ( y - &,k( s t + diam I j k s 3 diam(4,,)/2
(1.23)
349
1. Atomic Decomposition
(ii) The moments of aj,k coincide with those of $. (iii) We use the expression l l a j , k l l q = sup)jRa j , k ( X ) T ( X ) dxl, where the sup is taken over those functions with 1 ( ~ 1 1 S ~ , 1, l / q + l/q' = 1. So putting g(y, t ) = TJ * q t ( y ) we estimate I = ljq,ku(y, t ) g ( y , t ) / t dy dtl. BY(c) above we immediately get
6 2 1
( 1.24)
S ( x ) S ( g ) ( x )dx.
(R\Ek+i)n3Ij.k
Now, since S ( x ) 4 2k+1on R\Ek+l, and by (4.4) in Chapter XII,
(1.24) gives immediately I s C2k1h,k11/q, all j , k Clearly, the same estimate holds for I I a j , k l l q , and (iii) also holds. Next we indicate the minor modifications needed for arbitrary dimension n. The first change comes in the definition of 4, where we now let I ( y , t ) be the open cube centered at y of sidelength t ; this is a minor change. More important, however, we note that the decomposition of Ok = 4 , k given above is no longer valid. This is not a serious obstacle as we can use the Whitney decomposition 4.6 in Chapter XI11 instead. Let then { Q j , k ) be the Whitney'decomposition of o k and let 4 , k = interior of Qj&, all j, k T j , k and aj,k are defined exactly as before and the only property that has to be checked is (i) concerning the support of 4j.k. But as in (1.23) observe that if yj,k = center of h , k , then a j , k ( X ) # 0 implies there is ( y , t ) in T j , k such that
u
XIV. Hardy Spaces of Several Real Variables
3 50
X I s t and y E 4,k. Then Jx- y j k l s Jx- yl + Jy- Yj,kl s t + diam 4 , k s c dist(h,,, R"\Ok) + diam 4.k d c diam 4,k, since 4.k corresponds to a Whitney cube in the decomposition of 0,. This completes the proof. ly -
=
2. MAXIMAL FUNCTION CHARACTERIZATION OF HARDY SPACES In Chapter VII we considered the nontangential maximal function corresponding to a harmonic function in the disk and introduced the Hardy H P (T) spaces. Similar definitions hold for functions of several real variables. More precisely, given a function u(y, t) defined in R:+' we set Na(u)(x) =
a > 0.
SUP lu(y, r)l,
(2.1)
(y,t)Er.(X)
N. ( u ) is called the nontangential maximal function, of opening a, corresponding to u. We are interested in those classes of functions, or more generally, distributions, f which verify the following property: if u(y, t ) = f * P,(y) is the Poisson integral o f f , then Na(u) E L P ( R " )0 , < p < co. A straightforward argument shows that this last condition is independent of a (cf. 6.2 below), and for this reason we work with N , ( u ) = N ( u ) in what follows. A deeper fact, due to Fefferman-Stein [1972], is that the Poisson kernel above may be replaced by any sufficiently smooth function with nonvanishing integral and still obtain the same classes. Now suppose that a is a ( q, N) atom and is an integrable function, I D " + ( X ) ( ZG c(1 lxl)-( "+'+Iu! some E > 0, and la[s N + 1. If A(y, t ) = a * + t ( x ) , then an argument similar to, but simpler than, Proposition 1.2 gives that IIN(A)II, S clal, where c = cp is independent of a. For f E Y ( R " ) which verifies the assumptions of Theorem 1.10, let f = C aj(.Y') be the atomic decomposition given there and put F(y, t ) = f * + , ( y ) and Ai(y, t ) = aj * t,bt(y). We clearly have that N ( F ) ( X )s~1 N(A,)(x)" and also IIN(F)(Igs cClajp G cllS(v)ll$, c = cp independent off: In other words, the assumption S ( u ) E LP(R",0 < p < 1, implies that the nontangential maximal function associated to extensions off to R?+' by convolution with . about the dilations of appropriate functions also belongs to L P ( R " )What , the same true of S ( v ) ? the converse to this statement: If N ( F ) E L P ( R " )is A way to go about this question is to show that, if u is harmonic in R:+' and N ( u ) E L P ( R " ) ,0 < p d 1, then u = f * P,, where the distribution f admits an atomic decomposition; then apply Theorem 1.4. Although this approach works (cf. 6.5), we prefer to give a direct proof by means of a distribution function inequality we show first.
+
+
+
+
2. Maximal Function Characterization of Hardy Spaces
351
Lemma 2.1. Suppose u(y, t ) = f * P , ( y ) is the Poisson integral of an L2 function f and let V ( y ,t ) = ( t ( a / a t ) u ( y ,t ) , t Vu(y, t ) ) , where V = (d/ayl,. . . ,d/dy,). Then if EA = { N ( y )> A},
where c is independent of A and f:
Proof. Let OA = {N(,yEA* 6,)> f},where the infinitely differentiable function 6 has support in the unit ball and integral 1 and all its moments of first order vanish. Clearly, EAc OA a.e., and by a simple extension of Theorem 2.2 in Chapter VII loA/ < clEAl,with c independent of A. Let now (y, t ) E T(R"\OA) = U x c R n \ o , T(x). Then ly - < t for some x in R"\OA and consequently xEA* 4 , ( y )s f, or, equivalently,
XI
g(Y, t ) = x R " \ E A * +,(Y)
3
t-
(2.3)
Next consider
I =
I
S( V)(x)' dx
R"\c%
Observe that the innermost expression in (2.4) is bounded and that it does not vanish only for those (y, t ) in T(R"\OA). Therefore, by (2.3)
say. Since g s 1, by Proposition 3.4 in Chapter XII, J S cIlfll; < 00. We estimate J by considering separately the integrals corresponding to t ( a / d t ) u ( y , t ) = u ( y , t) and t V u ( y , t). We begin with the former. First, note that for 0 < F < 7 < co we have dt
352
XIV. Hardy Spaces of Several Real Variables
By Corollary 2.4 in Chapter IV, for almost every y E R", lims,o u(y, 6 ) =
f (y), lims+ov(y, 6) = 0, and lims,o g ( y , 6) = x ~ ~ , ~ ~Thus ( Y the ) . lower limit of the integrated term above goes to 0 with 6. As for the upper limit we need the estimates (2.6)
l U ( Y , t)l d C t - " ' P l l w ) l l p
Inequality (2.7) is a particular case of Proposition 2.3 in Chapter IV and (2.6) follows immediately from the estimate lub,
01
$5,
Nu),
(2.8)
where B ( y , t ) denotes the ball centered at y of radius t. Thus also limq,m u ( y , v ) v ( y ,v ) g ( y , 7)' = 0 and the integrated term vanishes. This gives at once that
-2
I
a
U(Y,
t)V(Y,M Y , t ) ;g(Y,
t ) dv dt.
R :+I
(2.9) Similarly, but integrating with respect to the space variables first and letting A denote the Laplacian in these variables, we get that
353
2. Maximal Function Characterization of Hardy Spaces
= J1
+ 52 + 3 3 ,
(2.11)
say. Since J2 and J3 are handled in a similar fashion we only estimate J2. Let T ( E A )= {(y, t ) E R:+’: d ( y , Rn\EA) < t } . T(EA) looks roughly like a collection of inverted cones based on the components of the open set EA. Now g ( y , t ) vanishes on T(EA) and so do all its derivatives. Furthermore, {(y, t ) E R?+’: ) u ( y , t ) l > A} G EA, and N Y , t)l
A
a.e. on
Rn\EA.
(2.12)
+
From the easily verified estimate *2ab s &a2 E - l b ’ , for arbitrary real numbers, it follows then at once that
E
> 0 and a, b
(2.13) The first summand in (2.13) does not exceed EJ.As for the second summand, by (2.12) it is bounded by &-‘A2
R:+’
( t $ g ( y , t ) ) 2 dy?.
Furthermore, since g ( y , t ) = 1 - xE, * + t ( y ) ,by Proposition 3.4 in Chapter XI1 this last expression is dominated by C E - ’ A ~ ~ 112’ ~ X=~CE-’A~~E,I , which is also a bound of the right order. In other words,
I J ~ ~ + I J ~ I s ~ E +J C E - ’ A ~ ~ E , I ,
E
> 0.
(2.14)
To complete the proof we only need to estimate J1;this integral looks like J2 but there are no derivatives acting on g. In first place note that
XZV. Hardy Spaces of Several Real Variables
3 54
On account of (2.6) limq+mu ( y , q ) g ( y , q ) = 0. Also since lirn6+,, g ( y , = ~ p \ ~ , ( y ) , it readily follows that Ilims+ou ( y , 6 ) 2 g ( y ,6)'l 6 N ( u ) ( Y ) ~ x ~ -and \ ~ ~consequently (~)
= 54
+ J5,
(2.15)
say. J,is all right. To bound J5we need another expression for ( a / a t ) g ( y ,t ) = -(a/at)(xE, * + l ) ( y ) . This is easiest obtained by taking Fourier transforms. Indeed, since (a/at)g(& t ) = -&(.f)(V+(t() * it readily follows that
e),
(2.16)
where qj is the C; ( R" ) function, supported in the unit ball, such that Gj(&) = (a/agi)&e). By the moment condition on clearly, q j ( y ) dy = 0, 1 S j s n. Returning to J5,by using (2.16) and integrating by parts we obtain that it equals
+,
= J6
I,.
(2.17)
37,
say. A moment's thought suffices to realize that J6 is a sum of integrals each of which is similar to J3, and consequently also
1 4 1 s EJ + C A ' ( EI.~
(2.18)
Moreover, since
2
+j = 1
xEA
* (?,)r(y)')xR:"\TIE,)(y,
t),
J, can be estimated as the second summand in (2.13), that is, by cA21EAl. Finally, combining (2.11), (2.14), (2.15), (2.18) and the above observations, and choosing E sufficiently small, we get that J s i J + c J R n , E , N ( u ) ( y ) ' dy + cA21EAl.But since J < CO, from (2.5) we obtain at once that Z S C J S C
I
R"\.%
N ( u ) ( y ) 2dy + cA21EAI.
(2.19)
2. Maximal Function Characterization of Hardy Spaces
355
We are now ready to prove Theorem 2.2. Suppose u(x, t) is harmonic in RI;+' and N ( u ) E L P ( R " ) , 0 < p < 2. Then if u(y, t) = t ( d / d t ) u ( y , f ) , also S ( u ) E L P ( R " )and there is a constant c = cp independent of u such that IIS(u)ll, s C ~ ~ N ( U ) ~ ~ ~ .
Proof. Assume first that u(y, t ) = f * P , ( y ) is the Poisson integral of an L2 function f: Then by multiplying (2.2) through by AP-' and integrating, it follows that
and by Theorem 3.3 in Chapter VII there is an L2 function f (which depends on E ) so that u(y, t + E ) = f * P , ( y ) . Therefore, by the first part of the theorem S ( t , (d/dt)f* P , ) ( x ) satisfies
XIV. Hardy Spaces of Several Real Variables
356
where c is independent of E. To complete the proof we observe that liminf,,, S(t(a/at)f*P , ) ( x )= S ( u ) ( x ) and invoke Fatou's lemma.
Remark 2.3. If u is harmonic in R:+' and ~up,,~j,~IU(y, t)("dy< 00 for some p > 0, then lim,,o u( t ) = f exists in the sense of distributions and u(y, t ) = f * P,(y)(Y').Indeed, as in Lemma 1.6 it is readily seen that also jRmIu(y, t)l dy d ct"-"'p, where c depends on u. Moreover, for any E > 0 the function u(y, t E ) is bounded and harmonic in R:+' and continuous in the closure of this set, and, consequently, u(y, t + E ) is the Poisson integral of u( , E ) . By taking Fourier transforms we have that I?(,$, t + E ) = I?(& E)e-r'*l and I?(,$, t ) = g([)e-'If' where a ,
+
-
Ig(5)l d erl*l I , i " ( y , t ) l dy s t"-"'Pe-'l*l.
Taking t = l/lSl it follows that g has tempered growth and consequently g E 9"(R " ) . Let f = inverse Fourier transform of g. Since limr+oG( -,t ) = g ( Y ) , we also have lim,,, u( t ) = f ( 9 " ) , as we wished to show. a ,
Remark 2.4. Combining the results of this chapter with those of Chapters IV and XI1 we have proved in particular that the following statements are equivalent forf E Y ( R " )and 0 < p < co: Let u(x, t ) = f * P , ( x ) denote the Poisson integral off and u(x, t ) = t ( a / a t ) u ( x , t ) . Then (i) limr,m u(x, t) = 0 and S ( u ) E L P ( R " ) . (ii) N ( u ) E LP(R").
-
Furthermore, IIS(u)ll, ~ ~ N ( and U )the ~ ~ constants ~ , involved in this norm equivalence are independent off: We introduce the Hardy spaces H P (R " ) of several real variables to consist of those tempered distributions f for which (ii) and consequently also (i) holds and set l l f l l H p = 11 N ( u)llp, 0 < p < CO. It is clear that for 1 < p < 00 the H P ( R " )spaces coincide with the usual Lebesgue L P ( R " )classes and that the norms are equivalent. The most interesting case occurs then when 0 < p d 1 and we pass on to consider some of the natural questions in this setting, including the relation to systems of conjugate harmonic functions, boundedness of multipliers, and interpolation.
3. SYSTEMS OF CONJUGATE FUNCHONS
+
Suppose ( n - l)/n < p < co. We say that the ( n 1)tuple of harmonic functions U = (u, u l , . . . , u,,) E HP(R:+') if it is a system of conjugate
357
3. Systems of Conjugate Functions
functions as defined in Section 3 of Chapter X, i.e., it verifies the generalized Cauchy-Riemann equations, and if in addition
We remark first that in this case u essentially determines U. In fact when p > 1 we observed that the uj's could be obtained from u by means of the Riesz transform, that u(x, t) = f * P,(x) and n
I
UIHP
IMIp
+
C IIRjfIIp - IIfIIp.
(3.2)
j=l
In this sense HP(R;+')can be identified with L P ( R " )What . is the situation for 0 < p s l ? The reason we restrict our discussion to the case p > (n - l ) / n is that the integrand IU(x, t)lp of (3.1) is subharmonic when p L ( n - l)/n. For the other values of p we must consider tensor functions of rank > 1, satisfying additional conditions. We do not discuss this general case and refer the reader to Fefferman-Stein [1972] where the full picture is presented. Note that if U E HP(R:+') a simple extension of Theorem 4.9 in Chapter VII gives that N ( u ) , N ( u , ) , . . . , N ( u , ) are in L P ( R " )and \lN(u)\\p,l \ N ( u l ) \ l p , *
* *
> IIN(un)llp
clulHp,
(3.3)
where c is independent of U. Our next observation is along the lines of Theorem 4.12 in Chapter VII.
Lemma 3.1. Suppose u is harmonic in R;+' and N ( u ) E L ( R " ) . Then u(x, t) = f * P,(x), where f and its Riesz transform RjJ 1 s j C n, are integrable. Furthermore, there is a constant c independent of u so that n
I I ~ I+I ~EIIRjfIIl 6 C I I N ( ~ ) I I ~ *
(3.4)
j=1
Proof. Since jRnIu(x,t)l dx s ~ ~ N ( by U the ) ~ analogue ~ ~ , of Theorem 3.4 in Chapter VII there is a finite measure p so that u (x, t) = p * P , ( x ) and lim,,,, u(x, t) = f ( x ) exists a.e. Moreover, since lf(x)I, I.(, t)(s N(u)(x), also lu(x, t) - f ( x ) l s 2 N ( u ) ( x )E L(R"), and, by the Lebesgue convergence theorem, lim,,ollu( -,t) -fIl1 = 0. This gives immediately that { u ( *, t)} is Cauchy in L(R")as t + 0 and consequently p actually coincides with the integrable function J: But this means that f~ H ' ( R " ) and by Theorem 1.10 there is a sequence { a k } of ( 1 , 2 , 0 ) atoms such that f = C a d y ' ) , Cbkl s ~ l l f l l ~=1cllN(u)ll1, and IlCc=, G - f l l ~ l + 0 as m + 00.
358
XIV. Hardy Spaces of Several Real Variables
In order to obtain (3.4), since the Riesz transforms are linear, we are then reduced to showing that for each such atom a,
IIRjaIIl s clal,
1d j
d
n,
(3.5)
where c is independent of a. This inequality looks like estimate (1.3) and is proved in a similar fashion. This is not surprising since both results involve estimating a singular integral operator. The details of this verification are therefore left for the reader. A more general result involving multipliers will be discussed in the next section. We are now ready to prove that (3.2) still holds for p S 1 provided we replace the last expression there by 11f l i p . For simplicity we do only the case p = 1, but clearly the argument extends to ( n - 1)/ n < p d 1 as well. Theorem 3.2. The following statements are equivalent (a) U = ( u , v l , . . . ,v,) E H'(R:+'). (b) There exists an integrable functionf, with integrable Riesz transform, sothat u = f * P t , v j = R j f * P , , lSjdn,andIUIH~-Ilf()~+C,n=,IIR,flll. (c) There is a distribution f so that u = f * P,,f E HI( R " ) ,vj = Rjf * P,, 1 d j s n and 1 ~ 1 ~ 111 fllH'. -
Proof. (a) implies (b). Since by (3.3) N ( u ) , N ( v l ) ,..., N ( v , ) are . . ,fn such integrable, by Lemma 3.1 there are integrable functions ifl,. that u = f * P,, vj = h * P,, 1 d j S n, and Rjf E L ( R " ) ,1 S j S n. Consider now the system of conjugate functions V = (f*P,, R 1f * P,, . . . , R J * Pt); clearly V E H'( R:+'). So U - V is also a system of conjugate functions in H'( R;+') and its first component is 0. Therefore by the generalized CauchyRiemann equations it follows that 5 * P , ( x ) - Rjf * P , ( x ) = cj, 1 S j s n, where cj is some constant. Furthermore, since I$ * P , ( x ) - Rjf * P,(x)(d c( Ilf;lll + IIRjflll)t-" + 0 as t + co,cj = 0, and U = V. The equivalence of the norms follows at once from Lemma 3.1 and an argument similar to Theorem 4.10 in Chapter VII. (b) implies (c). BY (3.3), N ( u ) E L ( R " )and IIN(u)lll = I l f l l ~ s ~ cIUIH,. The opposite inequality follows from Lemma 3.1. (c) implies (a). Because of Lemma 3.1 it is obvious. Remark 3.3. Theorem 3.2, in its version for ( n - 1)/ n < p s 1, contains the other half of the Burkholder-Gundy-Silverstein theorem. Indeed, if f E H P ( R " ) ,then, by Lemma 3.1, Rjf E L P ( R " ) ,and consequently by the implication (c) + (a), U = (f*P,, R l f * P , , . .. ,R J * P,) E HP(R:+'). For n = 1this covers the whole range 0 < p s 1, for n > 1 we must also consider Riesz transforms of higher order.
4. Multipliers
359 4. MULTIPLIERS
Theorems 5.1 and 4.5 in Chapter XI1 combine to give that if 0 < p s 1 and m satisfies a Honnander condition of order L > n / p and we denote this by m E M(2, L), then m is a bounded H P ( R " )multiplier. In other , by Theorem 1.10 words, the mapping T given by T f A ( ( )= m ( t ) - f ( ( ) which we may think to be originally defined for those f E CT(R " ) which are finite , c independent of sums of (p, 2, N) atoms, satisfies 11 TfllHps ~ l l f l l ~ pwith f; and consequently admits a bounded extension to H P ( R " )with the same bound. But this result is not sharp in the sense that we demand too many derivatives on the multiplier m. The atomic decomposition gives a more precise value of L and we discuss this next. We begin with some definitions. The notation k is reserved for the kernel obtained as the Fourier transform, in the sense of distributions, of m. First we consider what behavior of k is reflected from the M(2, L) condition on m. For this purpose we say that k verifies the G(2, L) condition, and write k E G(2, L ) , if
(1
RslxlsZR
1/2
IDBk(x)I2d x )
s cR-("/~+~~I)
for R > 0 and p any multi-index with IpI < L, and in addition if largest integer strictly less than L and L = + 3: then
(1
(4.1)
is the
(DBk(x) D P k ( x- y)(z dx
Rslxls2R
for all lyl < R / 2 , R > 0 and multi-indices p with IpI = i.The infimum over the constants c for which (4.1) and (4.2) hold is called the constant of the kernel k; similarly for the constant of m. A convenient notation is to denote by 1x1 R the annulus { a R s 1x1 s bR}, where 0 < a < b < 00 are fixed numbers which are unimportant in the conclusions. For instance, a = 1, b = 2 in (4.1) and (4.2). An important relation is given by
-
Lemma 4.1. Suppose m E M(2, L ) , L > n/2. Then k
E
G ( 2 , L - n/2).
Proof. We begin with some observations. Note first that, if 7 E 9 ' ( R"), then multiplication of either m or k by 7 only increases the constants of these
XIV. Hardy Spaces of Several Real Variables
360
new functions by c = c,. This is readily seen by using the product rule for differentiation and it is especially simple when q is supported in an annulus Ix,( R, which is the only case of concern to us. Also the conditions M and M are invariant under dilations of the form rn(5) + rn(t5) and k ( x ) + t - " k ( x / t ) , t > 0. Hence if rn satisfies an M condition, so does q ( t f ) r n ( S ) , with constant bounded uniformly in t. Now by the dilation invariance we may assume that R = 1 and show that the expressions on the right-hand side of (4.1) and (4.2) are finite. Let 4 be a nonnegative, Cm function, supported in {;< IyI < 2) so that C,"_-,4(2-'y) = 1 for y # 0. Such functions are easy to construct (cf. 6.1). Put now q ( 5 ) = 1 I,:, + ( T i t ) , and let rno(5) = q ( Z ) r n ( 5 )and mi(() = 4(2-'5)rn(t),i = 1 , 2 , . . . If ki denotes the distributional Fourier transform of mi, i = 0, 1 , .. ., we estimate first the expressions corresponding to the different ki's and then add them up.
-
.
Case i = 0. We estimate D B b ( xand ) DBki,(x) - DBb(x- y ) , which have and t8q(5)(1Fourier transform essentially equal to [@q(&)rn(@) eieY)rn((),respectively. First, since 0 < q < 1 and supp q E ((61d 2}, it readily follows that
with c = c., When estimating the difference we also have the factor ( 1 - e i * y ) in the Fourier transform side, and since ( 1 - eieYld 21y( on the support of q, we get that
Cave i
> 0. Since Clal=LI~OLI 2 c > 0 when 1x1
- 1, we get that
Furthermore, since t p d ( 5 )E Y ( R " ) , we see immediately that 2-'IS1tSrni(5) = (2-'5)B4(2-i5)rn(5) E M ( 2 , L ) , with constant bounded uniformly in i, and that this last function is supported 151 2'. Therefore, for
-
361
4. Multipliers
each a the corresponding term in the above sum is bounded by and we conclude that
(I,
lopki(x)l' dx)
1/2
C2i(n/2+I@I-L).
(4.5)
XI-1
The term involving the difference is estimated as before, and we get that
when IyI s 1. Notice that when ly12' > 1, we get a better estimate by using the triangle inequality and (4.5) instead. Now sum. From (4.3) and (4.5) we see immediately that
and this expression is finite provided that 1/31 < L - n/2, which is our assumption. Similarly, by adding (4.4) and (4.6), we observe that when I/?[ = L'= largest integer strictly less than L - n/2, the difference in (4.2) is bounded by clyl
+ clyl
1 "=1/Iyl
2i(1-y)+
1
2'y,
(4.7)
2'>1/lYl
where 0 < y = L - n/2 - L' s 1. It is now a simple matter to sum (4.7) and to note that for lyl s 1 it does not exceed c(yIy when 0 < y < 1 and clyl ln(2/lyl) when y = 1. We are now ready to consider the action of multiplier on atoms.
Lemma 4.2. Let 0 < p c 1 and m E M(2, L), where L > n(l/p - 1/2) 3 n/2. Suppose u is a ( p , 2, N) atom, supp a E I c B(x,, R ) , where R is of order sidelength of I and ~ I / 1 ~ p - 1S~21~1, 2 ~ ~and u ~for ~ 24 as in Lemma 4.1 set ki(x) = +(x/2'+'R)k(x), i = 1 , 2 , . . . . Then b i ( x )= k, * a ( x ) is also a ( p , 2, N )atom and lbil s C ~ - ~ ~ I U / where , c is independent of a, i, and E > 0. Proof. As it is readily seen that supp bi c B(x,, 2i+4R) and that the moments of bi coincide with those of u and thus vanish up to order N, it only remains to bound Ilbil12appropriately. First, note that by Lemma 4.1, ki E G(2, L - n/2), with constant bounded uniformly in i and R. Now let be the largest integer s N so that I? < L - n/2. Furthermore, let Ri(x,y) denote the remainder in the Taylor expansion of ki(x - y), as a function of y, about x, of order fi - 1. Then we can write Ri(x, y) as
*
r
XIV. Hardy Spaces of Several Real Variables
362
and ki(xo- y ) - Ri(x,y ) is a polynomial of degree at most k - 1 when considered as a function of y. Hence, by the moment condition on a, it follows that b i ( x ) actually equals
x { D ' ~ ( x- X O - S ( Y - x O ) ) - DPki(x- x O ) } dy ds.
(4.8)
Thus, to estimate 11 bill2,we may invoke Minkowski's inequality and consider the L2 norm of the expression in {. . .} in (4.8) above as a function of x for each 0 s s s 1 and y in supp a. Since sly - xol s R and Ix - xol S 2i+2R, is bounded by we see from the G ( 2 , L - n / 2 ) condition on k, that II{. .
i
c(2iR)n/~-n-A(sly - X012iR)L-n/2-A
,.(2iR)n/2-n- A
if L - n / 2 - 1 < k < L - n / 2 ,
' ~ ) - xol) (sly - ~ ~ 1 2ln(2'~/sly
,(2iR)n/~-n-G(sly - x 0 1 / 2 ' ~ )
if
if
k = L - n/2 - 1,
fi < L - n / 2 - 1.
The first two estimates are immediate and the third follows @omthe mean value theorem and the bounds for the derivatives of ki E M ( 2 , L - n / 2 ) . There are then three possible kinds of terms that will appear in estimating (4.8), one corresponding to each of the above expressions. Because all terms are handled in a similar fashion we only do one of them, the first one. In this case the corresponding expression in (4.8) is less than or equal to
c(~~+~~)-~(~/~-~/2)~-~(~-~(l/~-l/2))~n(l/~-~/2) 112.
Ila
In this case
E
= L - n ( l / p - 1 / 2 ) > 0 and Rn(l/P-l/2)
Ilal12~ c
[
~
~s clal. ~ ~
~
-
~
As the other two terms lead to similar expressions and since supp bi E open cube Q of sidelength of order 2'+4R,from the above estimate it follows that (bil s ~ Q ~ 1 ~ P - 1s~ 2 ~ ~ b i ~ ~ 2 S C~-~'IDI, and we are done. We also need Lemma 4.3. For rn, a, and as in Lemma 4.2 let 4 0 ( x )= 14 ( x / 2 ' + 'R ) and put b ( x ) = + o ( x ) k ( x ) Then . b o ( x )= b * a ( x ) is a ( p , 2, N) atom and lbol s cIaI, where c is independent of a.
~
~
5. Interpolation
363
Proof. Observe that supp bo E B ( x o ,24R) and that the support of the ki * a's is disjoint with that ball as long as i 3 4 . Since the moment condition is not disturbed by convolutions, it only remains to bound ~ ~ b o ~ ~ 2 appropriately, but this is not hard. First, by the above remarks it readily follows that lbo(x)l s ITu(x)l + C:=,lki * a ( x ) l , where T is the multiplier operator associated to in. Clearly, 11 Tall, s cllal12, and consequently (24R)n(l/P-1/2) 11 Ta 112 s clul. That a similar estimate holds for the other three summands follows at once from Lemma 4.2 and we have finished. We are now ready to prove
Theorem 4.4. Let 0 < p G 1 and rn E M ( 2 , L ) , where L > n ( l / p - 1/2). Then rn is a bounded multiplier on H P ( R " ) . Proof. Suppose first that f E H P ( R " )is a finite sum of ( p , 2, N) atoms, N > L, f ( x ) = I:=, a j ( x ) , say, so that Cjh=l\ajJ'5 cllfll"H. Then T(f) = C,h=, T (aj).Moreover, by Lemmas 4.2 and 4.3, and with a different decomposition for the kernel k adjusted to the support of each aj, we have that m ._
Taj(x)=
n . .
1ki * a j ( x ) = i=O
bi,j(x), i=O
say, where the bi,j's are ( 2 , p , N ) atoms and m
m
i=O
i=O
with c independent of j . Therefore, by Theorem 1.4, Taj E H P ( R " ) , IITaj((P,ps c(uj(p,and the same is true for Tf with h
IITTIIL~ s c CIujIp s C I I ~ I I L P . j=1
To obtain the same result for a general f in H P ( R " )we invoke Theorem 1.10 and 6.13 below. W
5. INTERPOLATION
We discuss one more application of the atomic decomposition, this time to a simple interpolation result especially suited to multipliers. We need a preliminary fact.
XIV. Hardy Spaces of Several Real Variables
364
Proposition 5.1. Let 0 < po s 1 < p < 2 and f E LP(R").Then given A > 0 we may write f ( x ) = X ( x )+ f A ( x ) ,where f A E HPo(R"), E L2(R") and
cAPocpIIfIIpP, independent off and A. IIfAIIzpo
with c
d
IIfAII2'S
cA2-PIIfIIpP,
hf. The proof is a slight variant of that of Theorem 1.10. Let v(y, t ) = t(a/at)f* p t ( y ) and for the given A put Ek = { ~ ( v>) A 2 k } and 0, = {M(xEt> ) $}. Here M is the Hardy maximal function defined with respect
to balls. Let = U jQ , k be a Whitney decomposition of 4, and 4 , k = interior of Qik. Then T j , k and aj,k(x)are defined as in Theorem 1.10 with N > n(l/po - 1); we claim that k=l j
k=-cn
j
will do. To compute the L2 norm of X we use duality. If 11q112= 1 and, with the notation of Theorem 1.10, g(y, t ) = 7 * $ t ( y ) , then
say. Clearly, K
11 [I2 d c. As for J, first, observe that
d c 7
J dc
I
s ( u ) ( x ) 2dx.
R"\E~
Indeed, if u,, denotes the volume of the unit ball, then
By definition, however, if (y, t ) E %, then I{x E R"\Eo: (y, t ) E T(x)}lt-"/v. > ;, and the last integral above is greater than or equal to 1.9. By the converse to Holder's inequality and (5.1) we get that r
as we wanted to show.
5. Interpolation
365
To estimate I l f A l l H ~ o , observe that if laj,kl denotes the atomic ( p o ,2, N) norm of aj,k,then as in (1.22) of Theorem 1.10 we have that
Furthermore, since when S ( u ) ( x ) / A> 1 we also have S ( U ) ( X = )~~ A ~ O ( S ( V ) ( X ) /sA Apo-pS(u)(x)p, )~O (5.2) and Theorem 1.4 yield llfA G C A ~ ~ - ~ I I S S( U~ )A~~ o~- ;~ I l f l l i .
llzpo
We are now ready to prove Theorem 5.2. Suppose T is a sublinear operator of weak-type (2,2) and bounded from HPo(R")into wk-LPo(R"), 0 < po 5 1. Then if po < p < 2, T maps H P ( R " )continuously into L P ( R " ) .
Proof. Suppose first p > 1. Given f E L P ( R " )and A > 0, write f = & + f A as in Lemma 5.1. Then {ITfl> A} c {IT&[ > A/2} u {ITf*l> A/2}, and consequently
1{1 Tfl > A l l G cA-*ll& 112' + cA-"0llfA
IIzPo
< ~ A - ~ A ~ - ~ I l f l l+ p P~ A - ~ o A ~ o - ~ I l f l l p= P ~A-~IlfllpP. Thus T is of weak-type (p, p) for 1 < p < 2 and also of type (p, p) in the same range of p's as a simple application of the Marcinkiewicz interpolation theorem gives. Consider next the case po < p G 1. We claim that if a is a ( p, 2, N) atom, then 11 Ta IIp s clal, where c is independent of a. Indeed, first observe that and there is an open cube Z so that supp a c I and I I a I I H p 0 s 21Z11/Po-1/Plal 1 1 ~ 1 s1 ~ 2 l ~ / ~ / ~ - ~Whence /~lal.
11~~11;
=.(I +I f0.r)
S c j L0.r)
)Ap-'l{lTal> AIldA
[r,m)
A P - l A - P o l l u I ( ~ PdA O +c
AP-1A-211al12'dA Ir.00)
s c( 111 l--PdPrP--Pola [Po + II 1 1 - 2 / ~ y ~ - 2 a1 1).'
(5.3) Setting r = in (5.3), we see immediately that IITaIIi G clap, as anticipated. Let n o w f e H P ( R " )be a finite sum 1 ui of (p, 2, N) atoms so
366
XIV. Hardy Spaces of Several Real Variables
that Clajlps 211f11;~. Since p s 1 we see that ITf(x)lps CITaj(x)lp and 1 1Tail/; s c Clajlp s c l l f l l % p . The general case folconsequently 11 ~ f l l ; s 1 lows easily by a simple limiting argument.
6. NOTES: FURTHER RESULTS AND PROBLEMS The classical theory of H P spaces is essentially part of complex analysis with many connections to harmonic functions and Fourier analysis. New methods are therefore needed to rid the theory of one-dimensional techniques such as conformal mapping, and extend the results to several dimensions. The recent n-dimensional real theory was started by Stein and Weiss [1960]; a crucial observation in this context is the fact that, if F = ( u o ,u l , . . . ,u,) is a (M. Riesz) system of conjugate functions, then IF(q is subharmonic for q > n/(n - 1). In the late 1960s important new developments took place, culminating in the Fefferman-Stein [ 19721 theory of H Pspaces of several real variables. We single out three such developments here. (i) Results concerning the boundedness of certain singular integral operators can be extended from Lp(R " ) , 1 < p < 00, to the H P (R " ) spaces, 0 < p s 1, and especially H ' ( R " ) . The methods used involve auxilliary functions such as the Lusin integral, a (vector-valued) singular integral itself. (ii) The result of Burkholder and Gundy and Silverstein[19711concerning the characterization of the Hardy space HP,(R:) in terms of nontangential maximal functions. This remarkable theorem, proved by probabilistic methods involving Brownian motion, raised many interesting questions, including the possibility of extending these results to R" and what role the Poisson kernel plays in all of this. (iii) Fefferman's identification of the dual of H ' ( R " ) with B M O ( R " ) [1971].
One of the main results of Fefferman and Stein is that the H P classes can be characterized without any recourse to conjugacy of harmonic functions or Poisson integrals. Elements u in H P ( R " )can be considered in terms of their boundary values f and have an intrinsic meaning: u is in H P ( R " )if and only if N ( f * + t ) ( x )E L P ( R " )whenever is a sufficiently smooth function, small at infinity, and has nonvanishing integral; in fact, it suffices to consider the radial maximal function N o ( f * &)(XI = sup,,olf* 44x11. In a different direction Calder6n and Torchinsky [ 19751established a similar characterization in terms of the Lusin integral S(f * $ , ) ( x ) corresponding to a smooth function $, small at infinity and with vanishing integral. The atomic decomposition is due to Coifman [1974] when n = 1 and to Latter
+
6. Notes; Further Results and Problems
367
[ 19781 for general n. Both of these results make use of the characterization of H P ( R " )in terms of maximal functions, and a relatively simple proof along these lines is discussed in 6.5 below. The atomic decomposition given here is based on ideas of Calder6n [ 1977b], Chang and R. Fefferman [1982], and, especially, Cohen [ 19821. Fefferman-Stein [ 19721, Burkholder and Gundy [ 19721, and Calder6n and Torchinsky [19751, considered the distribution function inequalities which allow for the control of the Lusin integral in terms of the nontangential maximal function, and vice uersa. The proof of Lemma 2.1 is based on some ideas of Merryfield [1985]. The results in Section 3 are due to Stein and Weiss [1960]; the work of Wheeden [1976] is also relevant here. The multiplier results in Section 4, which are due to Calder6n and Torchinsky [19771, can be extended in several directions (cf. Taibleson and Weiss [ 19801, for instance). The proof given here follows along the lines of Stromberg and Torchinsky [1980]; the sharp version of this result is due to Baernstein and Sawyer [1985]. The decomposition in Section 5 is essentially due to Chang and R. Fefferman [1982].
Further Results and Problems
There is a C:( R " ) function 4 supported in {fS 1x1 S 2) and such that 4(2-'x) = 1, x # 0. (Hint: If 7 is nonnegative, nonidentically 0, C,"(R") function supported in {fS 1x1 S 2}, then C,"_-, ~ ( 2 - j # ~0 ) for 77(2-'x).) x # 0. Look at 4 ( x ) = 7(x)/EJ:-, 6.2 Letf be defined on R:+' and suppose 0 < a < b < 00. Then I{ N b ( F ) > A}l d c(b/a)"l{N,(F) > A}I, all A > 0, where c is an absolute constant. (Hint: Let 0 = { N , ( F ) > A } and put 6, = {M,yo > ( a / ( a+ b))"}.As it is not hard to see that { N b ( F )> A} c_ 0,the conclusion follows at once from S the maximal theorem. Clearly, this result implies l[Nb(F)\lp c(b/a)"/PIIN,(F)IIp,0 < p < 00 (cf. Theorem 4.3 in Chapter XII).) 6.3 In case u is harmonic in R:+', then u E H P ( R " )if and only if the radial maximal function No(u ) E Lp(R") and 11 u 11 H~ 11 No(u ) I I p , 0 < p < 00. This statement corresponds to Proposition 5.5 in Chapter VII. The general result is the following: Let F(x, t ) be continuously differentiable with respect to the x variables in t > 0 and suppose that for some a, b > 0, N , ( F ) and Nb(lt V F I ) are in L P ( R " ) 0, < p < 0O. Then there is a constant c, depending VFI)(I"'(p+") on a, b, and p such that IIN,(F)IIPs cllNo(F)II~"P+"'IINb(~t if l l ~ o ( F ) l l p IINdlt VFI)IIp and l l W F ) l I p S ~llNo(F)llp otherwise. (Hint: On account of 6.2 it suffices to prove our assertion for a = 1, b = 2. = The desired inequalities follow without difficulty from the estimate I{ N ( F ) > A, N 2 ( ) V t F I ) S r-l'pA}l S cr-"/"I{No(F)> A/2}1 = I % ol , where r 6.1
El:-,
-
XIV. Hardy Spaces of Several Real Variables
368
is a number between 0 and 1. To show this estimate observe that if x E then there exists (y, t ) with ( x - y l < t and ( F ( z ,t ) l > h / 2 for ( y - zI s 4r””t. Thus, Q1 E {Mx%> (( r1lp/2)(1 r1/p/2))-1)”} and the estimate follows from the maximal theorem. The result is from the work of Calder6n and Torchinsky [ 19751.) 6.4 Suppose N ( F ) E L P ( R ” ) , 0 < p < m, and let N f ( F ) ( x )= SUP(y,:,)IF(Y, t)l(l + Ix -Yl/t)-”. Then IIN:(F)IlP s C l l N ( F ) I I P if A n / P andI{Nf(F) > t}l C ct-p(INf(F)II;ifh = n / p . ( H i n t : NotethatNT(F)(x)-c c sUPk 2 - * k ~ 2 k ( ~ ) ( which ~), in turn implies { N : ( F ) > t } c U z = 0 { N 2 k ( F> ) ~ 2 ” ~and t } then use 6.2.) 6.5 Suppose u(x, t ) = f * P,(x) is harmonic in R:+’ and N ( u ) E L P ( R ” ) , 0 < p c 1. Then f admits an atomic decomposition f = C a j ( Y ) into ( p , q, N) atoms and IIN(u)llg inf ClajIp,where the inf is taken over all possible decompositions. (Hint: The proof follows along the lines of Theorem 1.10 and is best understood when n = 1. By 6.2 also N,(u)E L”, and the open sets 0, = {N,(u) > 2k} = Uj 4,k, where the 4 , k ’ s are disjoint, T(4,k), where T ( I ) = {(y, t ) E R:: ( y open intervals. Let T ( 6 k ) = t, y + t ) c I}and put ?;;k = T(4,k)\T((!&+l)* Then
+
’
-
ui
f=xj j,k
u(y, t ) ~ l ( x - U ) d dt Yt=C(I”k(X),
?.k
i,k
say. As in Theorem 1.10 the proof is reduced to estimate
(1
?.k
1/2
(tlvub, t)l2 + rlv(y, t)l’)
dYdt)
*
To do this we invoke Green’s identity (4.5) in Chapter VII, which applies since the boundary aq,, is smooth enough for Green’s theorem to apply, and observe that the term in question is less than or equal to
Since we are working with the level sets for N2(u),and u is harmonic, it readily follows that lu(y, t ) ( , tlVu(y, t ) l , tlv(y, t)l s ~2~ on aq,, and the desired bound follows with no difficulty from this. This proof is due to Wilson [1985].) 6.6 To deal with operators acting on the Hardy spaces it is often necessary to work with sums of atoms, whose sumorts are stacked one on top of another. More precisely, we say that a function M ( x )is a ( p , q, N) molecule based at the ball B(xo,r ) , 0 < p s 1 < q < m, N > n ( q / p - l ) , provided it satisfies the following three conditions: 9 d crn(l-q’P), (i) ( R n l M ( x ) (dx
6. Notes; Further Results and Problems
369
(ii) jRn1M(x)141x - xolNdx s crN+"('-q/P), and (iii) j R n M ( x ) dx = 0 (this condition makes sense since by (i) and (ii), M is integrable). ), the uj's Show that if (i)-(iii) hold, then M ( x ) = C j u j ( x ) ( L q ( R " ) where are ( p , q, N) atoms supported in B(xo,2&+'r)and ~ ~ S cM ClujIp,~where ~ c depends only on the constants in (i) and (ii). This concept is due to Coifman and Weiss [19771 and is quite useful since it reduces the discussion of the continuity of many operators to showing that map atoms into molecules (cf. Theorem 4.4). 6.7 Show that, if, for 0 < p s 1, f E H P ( R " )n L ( R " ) ,then f ( x ) dx = 0. 6.8 Show that for each fixed ( x o ,t ) E R:+', P,(xo- x ) - P , ( x ) E H ' ( R " ) (as a function of x ) . Also iff is integrable and vanishes off a compact set K not containing the origin, then f~ H ' ( R " ) . (Hint: xK([)&/((Iis the Fourier transform of an integrable function, 1 S j S n.) 6.9 Suppose 4 verifies the assumptions of Z6's Theorem 2.2 in Chapter XI and f E H'(R"). Show that N , ( f * 4,) E L(R").It is also easy to see that, if $ = x1,I unit cube in R", and N ( f * $,) E L ( R " ) ,then f = 0, a.e. Along these lines Uchiyama and Wilson [1983] have shown that there is a nonnegative kernel 4 such that (0) f H b ( R ) = { f L~( R ) : N ( f * 4,) E L ( R ) } and H i ( R ) # H ' ( R ) . 6.10 There is yet another characterization of the H P spaces involving maximal functions. Suppose u is harmonic in R:+' and let u p ( x )= 11 t - ( " + ' ) / P U(y, t)llwk-LP(r(x),dydt), 0 < p < 00. Show that U E H P ( R " )if and only if up E L P ( R " )and I I u I I H p ~ ~ u p0 < ~~ p < p ,00. (Hint: One implica+" tion follows at once from the estimate I{(y, t ) E T ( x ) :( N ( ~ ) ( x ) / h ) ~ " "> t}l S c ( N ( u ) ( x ) / h ) ' :To prove the converse we show that N l 1 2 ( u ) ( xS) cup(x).Indeed, let ( y , t ) E r l I 2 ( x )and note there is a ball B centered at (y, t ) and of radius - t such that B c T ( x ) .Then by the Hardy-Littlewood Theorem 5.4 in Chapter VII and the estimate 7.5 in Chapter IV, for 0 < q < p we have
1
-
and the desired estimate follows easily from this. The result is due to Semmes [19831.) 6.11 (Hardy-Littlewood Imbedding Theorem) Suppose F is defined in R:+' and N ( F ) E L P ( R " ) 0, < p < 00. Then for p < q < 00,
~
P
370
XIK Hardy Spaces of Several Real Variables
the desired inequality follows readily from this by multiplying through by t"(q/P-')and integrating. This proof, as well as some applications, is in the work of Calder6n and Torchinsky [1975].) 6.12 The following extension of Paley's inequality, Theorem 1.3 in Chapter VI, holds:
(Hint: From 6.11, applied to F(x, t ) = f * ~ J ~ (and x ) p < q = 1, it ceadily / ~ - ' )the . mapping f + is follows that Ij(S)l ~ l l f l l ~ ~ I & I " ( 'Thus bounded from L2(R") into L2(R",de/le12")and from H P ( R " )into wkLP(R",d5/1[12n).By (a simple variant of) Theorem 5.2 this mapping is also continuous from H P ( R " )into LP(R",d[/1e12"),0 < p S 1, which is the desired conclusion. A direct proof using atoms also works.) 6.13 Endowed with the metric d(J; g ) = Ilf - gllfp, H P ( R " )is a complete metric space, 0 < p s 1. (Hint: It suffices to show that, if {fk} c H P ( R " ) and ~ ~ = l ~ 0,
OSm 2k}, 4 = { N ( x E k* 4,) > i},where 4 is a C m ( R " ) function supported in the unit ball with ihtegral 1, and put Ak = T (f!?k)\ T ( The following properties are then readily verified: Ak = RY+', and
u:=-*
1,.1
V(Y, t)12dY$S c
I
ok\
S( V)(x)' dx.
(2.3)
Ek+ I
Whence from (2.2) and (2.3) it follows that
k=-m
say. We examine each of the summands Jk in (2.4). By Proposition 1.2,
XK Carleson Measures
376
U ( y , t ) I 2 / t d y d t S cllfll',lt&l, and from the definition of the sets inOkl. consequently, volved it follows that jok,Ek+, s(V)(x)' d x s 22(k+1)l 1/2 k Jk ~ \ ~ . f ~ ~ lokk(1/2 * ~ ~ k C2klEkl ~ Ilfll*,and
2
k=-m
c m
Jk
cllfll*
2klEkl
cllfll*lls(v)IIl
c~~.f~~*~~g~~H1*
k=-m
Thus, by substituting in (2.4), we get ( L ( g ) lS Cllf(l*)lgllH1,as anticipated. To prove the converse, we use the characterization given in Theorem 3.2(b) in Chapter XIV and think of H ' ( R " ) as the subspace of L(R") consisting of those integrable functions with integrable Riesz transforms. Let B denote the Banach space consisting of the direct sum of n 1 copies ofL(R") normedby IlGll~ = ll(go, g 1 , . . ,gn)lls = C~=oIIgjIIl.ThenH'(R") can be identified with the closed subspace H of B consisting of those G's of the form ( g , R , g , . . . ,R,g), and by the Hahn-Banach theorem each bounded linear functional L,on H ' ( R " ) , or actually H, can be extended with no increment in its norm to a bounded linear functional, which we also denote by L, on B. Now the dual of B is essentially the direct sum of n + 1 copies of Lm(R") and consequently there exist L"(R") functions fo,. . . ,fn so that
+
UG) =
tJ
j=o
g i ( x ) ~ ( xdx, )
G E B,
(2.5)
R"
and ~ J " = o l l J ~ ~ m s norm of L. When restricted to those G's in H with g in C;(R"), the identity (2.5) reads
say. By Theorem 7.1 in Chapter XI, f~ BMO(R") and Ilfll, s c CJZollJllrn s c. norm of L. It thus only remains to show that f is uniquely determined by L. But this is not hard; first, the above representation is readily seen to hold whenever the integral converges and this is the case when g is, for instance, the H ' ( R " ) function P , ( x ) - P , ( x - y ) discussed in 6.8 in Chapter XIV. Whence, if I,. g ( x ) f ( x )dx = 0 for those g E H ' ( R " ) , it immediately follows that lim,+of* P , ( y ) =f(O) = c a.e. But constant functions f in BMO are actually (equivalent to) the 0 function and the uniqueness obtains. That the norm of L Ilfll* requires an easy argument using the first part of the proof. It seems natural to expect the atomic decomposition to play a role in the consideration of the dual to the Hardy spaces as well. In fact, even the
-
2. Duals of Hardy Spaces
377
proof of Theorem 2.1 can be simplified by invoking the atomic decomposition. We illustrate this in our next result.
Theorem 2.2. Suppose Lis bounded linear functional on H P ( R “ )0, < p =s 1. Then there exists a locally integrable function f such that
for every HP( R ” ) function g which is a finite linear combination of ( p , 2, N) atoms. Furthermore, f satisfies
where Pz (f)is a polynomial of degree s N = [n ( l / p - l)], and c is independent of the open cube I. The smallest constant c for which (2.7) holds for norm of L. Conversely, i f f verifies (2.7) and g is a finite every I is linear combination of ( p , 2, N) atoms, then the integral in (2.6) converges and IL(g)l s MllgllHP,where M the constant in (2.7) corresponding to$
-
-
Proof. We prove the second statement first. Let g = C J t laj be a finite sum of (p, 2, N) atoms, CJEllajlp Ilgll”H, and let 4 be open cubes containing lajl. j~~ Then 2 the support of aj with the property that & ~ 1 ’ p - 1 ’ 2 ~ ~ a
-
-
j=1
j=l
as we wanted to show. To discuss the representation of the functionals we start out by fixing an open cube I and consider the subspace H of L 2 ( I ) consisting of those functions with vanishing moments up to order N. Functions a in H belong 2 ~ ~ aif~ ~a 2E .H, then IL(a)l C to H P ( R “ ) and ))u)),P6 c ~ I ~ 1 ’ p - 1 ~ Thus c ~ I ~ 1 ~ p - 1and, ~ 2 by ~ ~the a ~Hahn-Banach ~2, theorem, L can be extended as a bounded linear functional to L 2 ( I ) with norm not exceeding C I I I ~ ’ ~ - ’ / ~ . By Proposition 3.2(ii) in Chapter I1 there is a function f E L2(I) such that llfll2 clI11’p-1’2and L ( g ) = JRnf(X)g(X)dx for g E L 2 ( I ) .Next we estimate ll(f- Pz(f))xz\\2,where Pz(f)is the polynomial of degree S N so
XV: Carleson Measures
378
that j r ( f ( x ) - P r ( f ) ( X ) ) X pdx = 0, IpI < N. For this purpose let h E L 2 ( I ) be a function in L 2 ( I ) with norm 1 so that Il(f- pI(f))xrII2s 2 l ( , ( f ( x ) - P I ( f ) ( x ) ) h ( x )dx(. Now, if P , ( h ) ( x ) denotes the polynomial of degree d N so that j r ( h ( x )- Pr(h)(X))Xpdx = 0 for S N, it is not hard to see that a ( x ) = ( h ( x ) - Pr(h)(x))Xr(x)is a ( p , 2 , N) atom with (a1S cl11 "P-1'2. Therefore,
which is equivalent to the estimate (2.7) for this particular cube I. It still remains to be shown that the functions which correspond to different cubes I can be thought of as restrictions to I of a single function f which verifies (2.7). But this is not hard to see since any two functions fi, X which correspond to cubes I, c 12, say, differ by a polynomial of degree d N on I, ,and consequently are actually (equivalent to) the 0 function in the space of functions which verify (2.7). Remark 2.3. The reader should compare the description of the dual of the Hardy spaces with the Lipschitz spaces introduced in Chapter VIII. This will be discussed further in 4.10-4.12.
3. TENT SPACES It is well known that the dual of the space of continuous functions on R:+' which vanish at infinity is the space of signed Bore1 measures on R:+'. It is therefore natural to consider whether the space of signed Carleson measures, i-e., those measures p on R;+' which verify
can be identified as the dual of some space of continuous functions on R:+'. For this purpose, and motivated by Theorem 1.1, let T = { f C(R:+'): ~ N ( f ) E L ( R " ) } ;endowed with the norm Ifl = IIN(f)11,, T becomes a Banach space. The triangle inequality is readily verified and the completeness follows without difficulty from the estimate (2.6) in Chapter
3. Tent Spaces
XIV,i.e., If(y,
379 t)l d
ct-"IIN(fllli = ct-"lfl. The statement we have in mind
is
Theorem 3.1. The dual of T is the space of (signed) Carleson measures in the following sense: if p is a Carleson measure, then
L(fl =
I
R:+'
f(Y, t ) M
(3.1)
Y , t)
-
is a bounded linear mapping on T with norm s c the Carleson constant of Ipl. Conversely, to every continuous linear functional L on T there corresponds a unique signed Carleson measure p so that (3.1) holds and the norm of L is comparable to the Carleson constant of p. Proof. The fact that every Carleson measure gives a bounded linear functional on T is contained in the inequality (1.1). The converse is not hard. That L must be given by a Bore1 measure p follows from the Riesz representation theorem. To show that p is actually a Carleson measure put f ( y , t) = ~ = ( ~ ) t); ( y fdoes , not belong to T but it is the limit, in the T norm, of functions in T. Since N ( f l = xB, by the continuity of L we get
Next, motivated by Proposition 1.2 we introduce the expression
where B runs over those balls which contain x, and define T" to be the class of g's for which C ( g )E L"(R"); the norm in this space is then lgl" = IlC(g)ll,. Can we identify T" as a dual Banach space? To answer this question we need one notation. Let T p = {f:S(f)E L P ( R " ) } 0 , < p c 00; the norm in Tpis Ifl, = IlS(fl11,. We then have
Theorem 3.2. Suppose f E T', g E T".Then
Thus each g E T" induces a continuous linear functional on T'. Proof. Let T h ( x )= {(y, t ) E R:+': Ix - yl < t < h } denote the cone with vertex at x truncated at height h, and set
XV. Carleson Measures
380
Note that S h ( f )increases with h and S " ( f ) = S(f).For g E T" we define the "stopping-time" h ( x ) as h ( x ) = sup{h > 0: S h ( g ) ( x ) M C ( g ) ( x ) } . Here M is a large dimensional constant to be chosen shortly. First, observe that, if B = B(x, h ) , then for some dimensional constants cl, c 3 1, we have
This is not hard to check; indeed, the left-hand side of (3.3) equals
as anticipated. Now, from this estimate it readily follows that if M is sufficiently large, then
I?: h ( z ) 2 h}l 3 lBl/2, all B. (3.4) = {z E B : h ( z ) < h} and observe that for z E Eh we have
l{z E
To see this let E h automatically that
and consequently by (3.3)
Thus lEhl S ( c 1 / M 2 )lcBl and IB\Ehl 3 IB1/2 provided M is sufficiently large, which is precisely (3.4). Finally, on account of (3.4), Fubini's theorem and Holder's inequality we get that
3. Tent Spaces
381 Sh(x’(fl(x)Sh(’”’(g)(x)dx zs c
I,.
S(f)(x)C(g)(x) dx,
(3.2) holds, and we have finished.
To complete the discussion of the duality of T’, as in the case of the Hardy spaces, we consider the notion of a TI atom. This is a function a(x, t ) supported in T ( B ) for some ball B c R “ so that dt JTiB/a(y,t)12 d q - = C < C O .
-
If we normalize a by setting
where the infimum is taken over all balls B for which supp a E T ( B ) ,then it follows at once that lal, = IIS(a)lll s c J Q J ,where c is a dimensional constant independent of a. Moreover, the following holds
Theorem 3.3. Every element f E T’ can be written as f = C j aj, where the aj’s are T’ atoms, and there is a constant c independent o f f so that Cjlajl
clfll.
Proof. The sketch of the proof follows. Let Ek = { S ( f l > 2k} and o;, = { M ( x E t )> y } , where 0 < y < 1 is chosen so that suppfE Uk T(ok). Furthermore, let { Q k j } be a Whitney decomposition of and for some large constant c let Bkj denote the ball concentric with Qkj that is c times its diameter. Then we can write T (t?k+l)\T( 4) as a disjoint union U jA, j , where Akj = T ( B k j )n ( Q k j x [ O , C O ) ) n (T(Ok)\T(Ok+l)), provided the constant c is sufficiently large. Now put a k j ( y , t ) = f(y, t ) X A k J ( Y ,t). It is clear that the asj’s are T’ atoms and that akj =f: To complete the proof we must then estimate I = Clakjl and show that it A} associate an open interval I, containing x such that (l/II,l) dp(y,t ) > A. Then 0,c I, and by passing to a disjoint sequence of intervals, (4)say, with the property that O,,E U24, the weak-type assertion follows. X )s~c inf, C ( p ) ‘ ,where Fix now I; we must show that (1/111) I,C ( ~ ) ( dx c is independent of I. To do this for each x in Z divide those open intervals Q containing x into two families by setting = {Q:IQI S 1211) and $2 = { Q :1 91> 1211) and proceed as in Proposition 3.3 in Chapter IX. This observation is due to Deng [1984].) 4.6 The following extension of Theorem 3.2 holds: given f defined in R:+’, let
IT(,,)
uxsoA
Then
+
where l/p l / p ’ = 1, 1 c p s 00, and c is independent of f and g. (Hint: We do the case n = 1, p < a.Let Ek = { A p ( f > ) 2k} = Qj,k, where the Q’S are disjoint, open intervals and 0 k = { M ( x E k )> 41 = Uj 4,k, where the Z’s are also disjoint open intervals. Observe first that
uj
c
J
C(lglP’)(x)”P‘dx s c
J
C(lglP’)(x)’’P’ dx, Ek
where c is independent of k. This follows immediately from the chain of
386
XV. Carleson Measures
inequalities
here we used the fact that C(lgl”)(x)””‘EA , . Let now %j,k = T ( z j , k ) \ u m T (z m , k + l ) , q , k = 4,k\U,,, J m . k + ’ . Then by Holder’s inequality we have
say. Furthermore, since
and
we also have
This result is also from Deng’s work [1984], and it can be used to give the following interesting extension of (1.1). Let t,h be a Littlewood-Paley function and suppose g E B M O ( R ) .Then by Corollary 1.3, lg * t,h,(x)I’/tdxdt
387
4. Notes; Further Results and Problems
is a Carleson measure for q Z 2, and A2(f* & ) ( x ) = S(f* A ) ( X verifies ) S cllfllp, 1 < p < co. Thus for 1 < p < 2 we have IIA2(f*
provided that 277/(2 - p ) 3 2, Le., 77 2 2 - p ; clearly, (4.1) holds also for p 3 2, 7 3 0. In particular if 7 = 1 we get (4.2)
which is not a consequence of (l.l),since as we have seen in Problem 4.3 lg * t,bt(x)1/ t dx dt is not necessarily a Carleson measure. Can (4.2) be extended to the case 0 < p S 1 ?) 4.7 Coifman and Weiss [19771 observed that HI( R") is a dual space; more precisely, it is the dual of VMO(R") in the sense that each continuous linear functional L on VMO(R") has the form L ( f ) = J R n f ( x ) g ( xdx, ) f~ Co(R")n VMO(R"),g E H ' ( R " ) and the norm of L ((g1IH1. 4.8 A Carleson measure p with the property that limlBl+op( T(B))/IB= ( 0, is called a vanishing Carleson measure. If the mapping f + f * PI is compact from L P ( R " )into LP,(R:+'), 1 < p < 00, then p is a vanishing Carleson measure. How about a converse? (Hint: Suppose not. Then there exist E > 0 and a sequence {&} such that l&l+ 0 and p ( T ( & ) ) / l & l Z E. By Proposition 3.2 in Chapter I1 we can find a subsequence, which we denote by {Bk}again, and f E L P ( R " )so that ( & - l ' P X B k converges weakly to f in L P ( R" )We . claim that f = 0 a.e. Indeed, let EA = {f> A} and set g = X E , ; since f~ L P ( R " )it is easy to see that g E Lp'(R"),l / p + l/p' = 1. Thus, on the one hand,
-
lim ~ ~ " 1 4 i " ' x B k ( X ) g (dx x )= j R n f ( x ) g ( xdx )
k-tm
and on the other hand
as k + 00. Consequently j { f > A ) f ( xdx) = 0 for each A and f = 0 a.e. Since P , ( y ) E LP'(R"), we also get that limk+oolBk(-l'P,yB, * P t ( x ) = 0 everywhere, and by the compactness assumption I((&l-''pxBk * ptllP+ o as k + 00; this contradicts the fact that p ( T (&))/ I Bkl 5 E. The result, as well as the answer concerning the converse, is in the work of Power [1980].)
XV. Carleson Measures
388
4.9 Theorem 2.1 establishes that, given f E BMO( R"), we can find n + 1 bounded functions fo, fl ,. . . ,fn so that f = fo - x,n=l Rjfi. In this direction Uchiyama [1983] has proved the following result: Given O , ( [ ' ) , . . . ,en([') E C"(H), let ( K j f ) * ( 5 )= 8j(S/l&l)f(S),1 S j S n. It is well known, and not hard to see, that there are n scalars aj and smooth functions Ckj homogeneous of degree 0 such that
whenever f~ L P ( R " ) ,1 < p < 00, say. For f E BMO(R") the definition must be modified as in Theorem 7.1 in Chapter XI. If
then for any f E B M O ( R " ) with compact support there exist g l , . . . ,g, L"(R") so that
c m
f=
E
rn
K,gj
(modulo constants),
j=1
and
c l l g j l ( m=S cllfll*, j=1
where c depends on the 8's but is independent of J: Since
Uchiyama's result includes the assertion of Theorem 2.1 alluded to in problem 4.9. By duality arguments one can also show l l f l l ~ 1 C,?,lIKjflll. 4.10 We say that a measure p in R:+' is a Carleson measure of order p2lif
-
p ( T ( 6 ) )6 clOlp,
all open sets 6 = R".
(4.3)
A simple argument, using a Whitney decomposition, gives that it is sufficient to check (4.3) for cubes. For 0 < p < 1, let m 5 1 be the smallest integer >n[l/p - 11. Then for harmonic u, u + limt+ojRnu ( x , t ) g ( x ) dx gives a continuous linear functional on H P ( R " ) provided that d p ( x , t) = It"(a/at)"'(g * P , ) ( x ) l * / t d x d t is a Carleson measure of order p. (Hint: The expression in question equals
I =c R:+'
t m ( $ ) r n u ( ~t,) t r n ( $ ) r n ( g * P , ) ( x ) dx-.dt
t'
from this point on the proof proceeds along the lines of Theorem 2.1. I can
4. Notes; Further Results and Problems
389
also be estimated by making use of 6.11 in Chapter XIV if one assumes instead that I t"(a/dt)"(g * P , ) ( x ) (S c t " ( l / P -.)l ) if 4.11 Let 0 s a,k = [a].We say that f~ ' i P q ( R " )=
Ifla,¶ = sup( t-"-'lq inf PEP,,
I
I f ( y ) - P ( y , x, t)lqdy)'lg < 00,
Ix-ylsr
where the sup is taken over x 6 R", t > 0, and Pkis the class of polynomials of degree s k This expression is a seminorm in $"*¶ and Iflm,¶ = 0 when f is a polynomial of degree s k; becomes a Banach space when such polynomials are identified. We also say that a harmonic function is in H".¶(R:+') = H a 7 4 if
then where the sup is taken over (x, t ) E R:+'. Prove that if f~ and I u ~ , , ~ S clfla,¶. Conversely, if u E u ( x , t ) = f * P t ( x )E f = limt+ou( t ) exists, it is in 8"*¶and verifies Ifla.,¶s C ( U [ , , ~ .When q = 2 we are dealing with Carleson measures of order p 3 0. These "trace" results where considered by Fabes and Johnson and Neri [1976] and Ortiz and Torchinsky [ 19771. 4.12 In Theorem 2.2 we may use (p, q, N ) atoms, 1 < q < a,instead of ( p , 2, N ) atoms. This gives automatically an equivalence of norms in the spirit of Theorem 5.1 in Chapter VIII. 4.13 Theorem 3.3 holds for T p = {f:I f l , = IIS(f)llp 0. We then have that, if p E C m ( v ) , then p ( { M f ( x ,t ) > A}) G c1 jIMf,c2A) u ( x ) dx. This result in particular implies that for u E A", 1 < p < 00, then p E C,(u) if and only if j R : + l Mf(x, t)" d p ( x , t ) G c jRnlf(x)Ipd v ( x ) . As for the general weak-type result, they show that Mf(x, t ) maps LP,(R")into wk-LP,(R:+'), 1 G p < 00, if and only if p E Cp(u).When d p ( x ) = w ( x ) dx, this statement coincides with statement (7.3) in Chapter IX. To state the strong-type results we need one more notation: we say that p satisfies the F,( v ) condition, 1 < p < 00, and write p E Fp(u ) provided that
M(V'-"'XI)(X,t ) , dp(x, t ) s c
I,
v(x)-"("-') dx < 00,
for all open cubes I. Then for 1 < p < 00 the following holds: Mf(x, t) maps LP,(R") continuously into LP,(R:+') if and only if p E Fp(u).The proof follows along the lines of Theorem 4.1 in Chapter IX. 4.16 It is also possible to establish weighted and vector-valued inequalities in this context. We describe briefly a couple of examples from the work of Ruiz-Torrea [1985b].Consider then the maximal function Mf(x, t ) of 4.15 and the fractional maximal function M,,f(x, t) given by
where the sup is taken over those cubes I centered at x of sidelength at least t. The vector-valued estimates that hold are of the type
11 11 TfjlllrllLq(R:+l) cII ~
~ ~ ~ ~ ! r ~ ~ L p ~ R n (4.5) )
391
4. Notes; Further Results and Problems
and P({llTJlll~> A})
cA-¶Il
II~II1'114L~R").
(4.6)
In case Tf(x, t) = Mf(x, t ) , (4.5) holds with 1 < p = q < co, and (4.6) with q = 1, provided p is a Carleson measure. In case Tf(x, t ) = M,f(x, t ) and p = s(1 - v / n ) 3 1, (4.5) holds for l/p = v / n + s(1 - v / n ) / q ,s < q < a, and (4.6) with q = s, provided p is a Carleson measure of order p. 4.17 Suppose that a E BMO(R"), i,b is a smooth Littlewood-Paley function, and w E A2. Then d p ( x , t ) = la * i,bt(x)12w(x)d x d t l t is a Carleson measure in C,(w) with constant Sclla11;, where c depends on $ and w but is independent in A2. (Hint: Repeat the proof of Proposition 1.2 carefully; this result is due to JournC [19831.) 4.18 Assume that d v ( x ) = u ( x ) dx, u E Ap,1 < p < co,and 4, $ are smooth functions with I,. $ ( x ) dx = 0. Show that the operator Tf = (Ilo,oo)lf* 4,l21a* $#t d t ) ' I 2 is bounded in LP,(R").(Hint: Suppose p = 2 first; 4.17 gives at once that T is bounded in L$(R")with constant independent in A2. The general result follows by theorem 7.6 in Chapter IX. This observation is also due to JournC.)
CHAPTER
XVI Cauchy Integrals on Lipschitz Curves
1. CAUCHY INTEGRALS ON LIPSCHITZ CURVES
Suppose r is a curve in the complex plane C given by z ( x ) = x + i+(x), x E R ; our only assumption is that +‘ E L“(R). As in Section 3 of Chapter X we are interested in the following question: given a continuous, bounded function f on r, does there exist a function F ( z ) , analytic in C\r, so that lim,,o+ F ( z ( x ) + i q ) - F ( z ( x ) - i v ) = f ( z ( x ) ) ,x E R? Our first approach is to consider the Cauchy integral off on r, that is to say the function
Suppose first 77 > 0, we then have
=Z+J,
say. Now a straightforward computation using residues gives
392
1. Cauchy Integrals on Lipschitz Curves
393
and consequently 1irnq+, J = f ( z ( x ) ) / 2 On . the other hand, formally at least,
A similar argument works for 7 < 0 as well. It becomes thus apparent that it is imporant to study the operator
i.e., the singular integral with kernel k,(x, y) = ( l / ( ( x- y ) + i ( 4 ( x )d ( y ) ) ) ;the factor 1 + i+'(y) may be omitted since 4' E L". So, before discussing the various properties of F ( z ) we introduce the singular integral operator
C,f(X)
= P.".
J
k,(x, Y ) f ( Y ) dY,
fE C3R).
(1.1)
R
Then we consider, as we did for the case 4(x) = x, i.e., the Hilbert transform, the questions of pointwise and norm convergence and continuity of CJ in the various Lp(R ) spaces. Although these problems are easily formulated, they are rather difficult to solve. A way to go about this is to suppose that 4 has Lipschitz constant x, > y and we have to x > x1 >
- -
r ( x - XI, t )
* *
r(x, - y, t ) = Jtj-"r(x- y, t )
(1.13)
there. If on the other hand t < 0, then D ( x , y , t ) becomes x < x1 < . * < x, < y, and the identity (1.13) still holds. Whence, in either case (1.12)equals
I tl-"r(x - y , t )
I
D( X.Y, 1)
$(x,)
. . $(x,,) dx, . . dx,. *
*
(1.14)
XVI. Cauchy Integrals on Lipschitz Curves
400
To compute the integral in (1.14), we observe first that if I is the interval (min(x, y), max(x,y)), then D(x, y, t ) c I x - x I. Furthermore, for u E S,,, the permutation group on the set of n elements, the sets u(D(x, y, t ) ) form a measurable partition of I x * x I, where u acts on R" by exchanging the coordinates. Therefore the integral in question verifies
-
+
--
n!j
+(XI)* * * +(Xn) dx1. * * dxn D(x,w)
=
(I,
+(XI)dx,)
* * *
(I,
+(Xn) dxn)
= (sgn(x - Y))"(4(X) - 4(Y))",
and the kernel of R,(M,R,)" is then (l/n!)(sgn(x - Y ) ) " ( ~ ( x-) 4(y))"ltl-"r(x, y, t). It is now easy to complete our proof, for we must only evaluate
and we are done. W Proposition 1.5 asserts, then, that the mapping
is bounded in L2(R). To establish the continuity of Calder6n's commutators of higher order, we pass to consider the other ingredients in the proof of Propositions 1.3 and 1.5. We begin by introducing the Littlewood-Paley
function
and prove a preliminary result that will enable us to estimate the dependence of IIGJII, on n.
Lemma 1.7.
For each t > 0, I(MqJ',)"f(X)Js. 11+1(2Mf(x),c indep. of n.
-
Proof. Clearly I(M,P,)"f(x)( s (($(lL(k,* . * k , ) * (fl(x). Now, k is positive, even, decreasing in [0, m), and has integral 1, and the same is true of 9
1. Cauchy Integrals on Lipschitz Curves
k*
+
- * k. Thus by Proposition 2.3 in
( k*. *
*
Chapter IV, (k,*
* k ) , * I f l ( x ) s c M f ( x ) ,and we have finished.
40 1
-
* *
* k,)* Ifl(x) =
We can now prove
Proposition 1.8. Suppose
+
E
L"(R"), 11 +llmG 1. Then (1.15)
where c is independent of n,
+, and$
Proof. The case n = 0 of (1.15) is (1.3) and the case n = 1 is inequality (l. l l ) , which was proved by combining (1.3) and Lemma 1.4; this suggests that an induction argument along those lines might work. Suppose then that (1.15) holds for 0 s k S n, and consider G,+,J: Putf.,, = (M,P,)"fand observe that G , + , f ( x ) = (jIo,m)l Q,M,P,Jn,,(x)12/tdt)"'. We invoke Lemma 1.4 for each fixed t and then Minkowski's inequality and we see at once that
) l , by say. To estimate A we observe first that, if F ( x ) = ~ u p , , ~ l f ~ . , ( xthen Lemma 1.7, F ( x ) < , M f ( x ) and consequently llF211s cllfl12,with c independent of n and J: Furthermore, since IP,J,,,(y)Js , P , F ( x ) for ( y , t ) E T ( x ) , we also have N ( P J n . , ) ( x ) s , M F ( x )and by Theorem 1.1 in Chapter XV, A s cll$llrnllN(l',f,,,)ll, s cllMF112 s cUl12, with c independent of n and J: A similar argument works for B as well. Indeed, by (1.4) and since 1q(x, t)l = kl,l(x), it readily follows that
again with c independent of n and$ It only remains to estimate C, but this is easy since on account of (1.4), applied twice, and the induction hypothesis it readily follows that
XVZ. Cauchy Integrals on Lipschitz Curves
402
where c is an absolute constant. Whence adding the bounds for A, B, and C we get IIGn+lfl12 c(n + 1)llfIl2 + cllfl12 C c(n + 2)Ilfl(2, and the proof is complete. We are now ready to prove one of the basic results in this chapter, namely, the boundedness of Calderh's commutators of higher order, with an appropriate control on the norms.
Theorem 1.9. Suppose 4 is a Lipschitz function on R, f c C Z R ) Put
Il+'lloo~1, and for
Then there is a constant c, independent of n, 4, and f, so that IICJII2
(1.16)
s c(n + 1)"llfI12.
Proof. Let JI = 4'. By Proposition 1.6 it suffices to prove (1.16) with p.v. j R R , ( M , R , ) " / t f d t in the left-hand side there. To study this operator M,(P, - iQ,) into we expand first R,(M,R,)" = ( P , - iQ,)M,(P, - iQ,) 2"+' terms of the form ToM,T, * * * M,T,, where is either P, or -iQ,, 0 =sj c n. Among these terms there is exactly one with = P,for allj, and since P, is even its p.v. is 0. In addition there are (n + 1) terms where Q, appears exactly once, and in the remaining terms Q, appears at least twice. We discuss first those terms where Q, appears at least twice. There are n terms where To= -iQ, and there are also those terms where -iQ, appears for the first time in place of Tk and for the last time in place of T,, 1 s k < rn C n. Because both cases are handled in a similar fashion, we only discuss the latter situation here. The typical expression we consider, then, is given by the string
---
P,(M,P,)~-'M,Q,(M,R,)"-~-'M,Q,(M,P,)"-". (1.17) First observe that the p.v. integral corresponding to (1.17) can be written jectc1,& + lim&+o j-lleO
II,
N X , Y l f ( Y ) dYl.
X-Y~>E
Then T* is bounded in L P ( R ) ,1 < p < 00, and its norm depends only on IlA’llm and JIB’JJm. The proof follows essentially along the lines of Theorem 1.10 and is therefore omitted. Remark 2.4. The dependence of the norm of T* on llA’ll, and IIB’llm is important in applications. From estimate (2.3) and the proof of the BenedekCalder6n-Panzone principle, it follows that the norm goes to 0 with IIB‘llm. Also, if F ( 0 ) = 0 it is not hard to see that the norm goes to 0 with llA’llm.
We also point out an n-dimensional result. Theorem 2.5. Let A, B be (complex valued) Lipschitz functions on R“, and let M ( x , y ) denote the singular kernel
Furthermore, let T , f ( x ) = jlx-y,,e M ( x , y)f(y) dy, and put T * f ( x ) = supe,,,lTJ(x)I. Then T* is bounded in L P ( R ” ) 1 , < p < 00, and satisfies IIT*fll, s cllfllp, where c depends on the Lipschitz constants of A and B and p but is otherwise independent of A, B, and f:
2. Related Operators
41 1
Proof. We invoke the method of rotations, Theorem 3.1 in Chapter X. First observe that
[
( M ( x ,x + Y l f ( X 2 lYl>E which in polar coordinates becomes T&f(X)=
2
P
I
+ y ) + M ( x , x - Ylf(X
( M ( x ,x + uy')f(x
- y ) ) dy,
+ UY')
( E P )
+ M ( x , x - uy')f(x - uy'))u"-'
du dy'
say. Write now x in R" (uniquely) as x = w + ty', where y' is a fixed vector in X, t E R and w E the hyperplane orthogonal to y' which passes through the origin. With this notation we have
T&,ff(X) =
TE,ff(W
+ tY')
=
I
M(w + ty', w + uy')Jt
IU--II>E
- uJ"-'f(w
+ uy') du.
It is clear that in our case
+ ty', w + uy')lt - u y - B( w + ty') - B( w + uy')
M(w
( t - u)'
A( w + ty') - A( w + UY')
'
))
--(n+1)/2
' which for each fixed y' E Z and w E R" is one of the operators covered by there. By that result it follows Theorem 2.3 with F ( z ) = 1/(1 z')("+')/' that for 1 < p < 00, t-u
+
where c is a constant which depends solely on the Lipschitz norm of A and Byand p. Thus,by Minkowski's inequality and Fubini's theorem, we finally get
XVI. Cauchy Integrals on Lipschitz Curves
412
3. THE T1 THEOREM For many operators in analysis an important question is to decide whether they are bounded in L2. In this section we discuss a simple criteria for this to occur. Suppose T is a linear operator, which is continuous from the Schwartz class Y ( R " )into Y'(R").As in 8.12 in Chapter XI, we assume that there are a kernel k(x, y ) defined for x # y in R" and constants c and 0 < 6 S 1 such that the following three properties hold: (i) I@, Y)l C I X - Yl-", (ii) for all xo, xy y in R" such that Jxo; XI < Ix - y)/2, J k ( x oy, ) -("+a) k(x, u)l + M Y , xo) - k(YY x)l 4x0 - X I S I X - Yl (iii) for each pair f, 9, of disjointly supported, C,"(R")functions, the evaluation of the distribution Tf on the test function 4 is given by Tf(4 ) = .fRn J R n y ) f ( y ) 4 ( x )dy dx* 9
k(x3
As in Chapter XI, T is called a Calder6n-Zygmund operator if it can be extended to a bounded operator in L2(R"). It is clear that the adjoint T* of T, defined by T * f ( + ) = T4(f),is associated with a kernel h(x, y) which verifies the same properties as k; in fact, h(x, y) = k ( y , x ) . Observe that it is possible to define T1, the image of the function identically 1 under T; T1 will be a distribution on those test functions in C:(R") with vanishing integral. In fact, let f~ L m ( R " )n C"(R") and 4 E CF(R")have integral 0; we want to define Tf(+).Let fl be a C:(R") function which coincides with f on the support of 4, and put f2 = f - fi . T'(4) is well defined, and in analogy with (iii) above we give a meaning to T f , ( 4 )= J R " J R n k ( x , y ) f i ( y ) 4 ( x )dy dx. Let xo E supp 4, and note that since $C has integral 0, by integrating first with respect to x we obtain
where c depends on 4. Whence
and Tf,(4) makes sense. Since it is clear that Tfl( 4 ) + 7''.4 ) is independent of the choice of the decomposition fl +f2 off, we define T f ( 4 )to be this value.
3. The TI Theorem
413
In order to state the desired result we need a couple of observations. T1 E BMO means that for all 4 E CF( R") with integral 0 we have I T1(4)1S cII 4 11 where the constant c is independent of 4. Also if for a function $ on R" we let $:(x) = t-"t+h((x - z ) / t ) ,then we say that T has the weak boundedness property if for any bounded set B c C r ( R " )there exists a constant c which depends only on B so that for all 4, t+h in B, x in R" and t > 0, IT$:( +:)I s ct-". We are now ready for Theorem 3.1 (David-JournB). Let T be a linear operator which is continuous from Y ( R " ) into Y'(R"),and assume that it verifies properties (i)-(iii). Then T can be extended to a Calder6n-Zygmund operator (bounded in L Z ( R " ) )if and only if the following three conditions are satisfied:
T1 E BMO T*l
E
(3.1)
BMO
(3.2)
T has the weak boundedness property
(3.3)
Proof. That the conditions (3.1) and (3.2) are necessary follows immediately from 8.14 and 8.15 in Chapter XI and (3.3) follows from Holder's inequality. To show that the conditions are also sufficient we first describe a (weak) representation formula for Tf for smooth f s , and then proceed along the lines of Proposition 1.5. Let 4 be an even, nonnegative, C r ( R " )function with integral 1, and to conform with the usual notation, let P, denote the operator convolution i.e., P J ( x ) = f * &(x). Let f E CF(R") and observe that since with
+,,
-(P:TP:) a = (&P:) TP:+ P : T ( $ P : ) , at
then Tf = -1im P : T P : f l f / " = -1im
I .+' E+O
=
-1im
I
'+'
(E,l/E)
a
dt ( P :TP:f)at t
t-
t ( i P : ) T P : f dt y
(E,l/E)
say. This is the representation formula alluded to above. Since the expressions A and B correspond to operators which are the adjoint of one
414
XVZ. Cauchy Zntegrals on Lipschitz Curves
another, it suffices to estimate one of them, A say. We begin by taking a closer look at t((a/at)P:). First observe that
= 2( j = 1
j,ctt)Gj(tt))s^tn,
say. Note that according to our assumptions on 4, Jj(0) = Gj(0) = 0 , l S j S n. Thus if and q denote the vectors (+l,. . ,+,,) and ( T ~. ., . ,T,), respectively, and Q+,ry d Q,,, dentte the vector valued operators defined by (Q+,tg)( 6 ) = t X E ) + ( f & ) , (Q,,tg) (5) = g ( 5 ) 6 ( t S ) , then t((alat)P:)g = 2Q+,,* Q,,rg. Returning to our representation, and with the notation M, = Q,,,TP,, we study the operator given by
+
.
To estimate (3.4) it is convenient to have the integral representation of M,g at hand; as is easily seen it is given by J R n K ( x , y, t ) g ( y ) dy, where K ( x , y, t ) is the n-vector with components pj(x,y, t) = P(qj);(r#4), 1 S j S n. Now, from properties (ii) and (3.3) it readily follows that for some 0 < 8’ < S and l s j s n , (3.5) where c is an absolute constant which depends only on 4 and q. Estimate (3.5) allows us to compute M,1; indeed it is the n-vector given by the absolutely convergent integrals
I,”
pj(x, Y, t ) dY = T*(qj);(l) = Tl((qj);)
*
= ( T l ) ( V ~ ) ~ ( X ) ,1 d j
n.
(3.6)
In particular, since by (3.1) T1 E BMO and the qi’s are Littlewood-Paley functions, IM, 1 1/ t) dx dt is a Carleson measure with constant S c 11 T1 I(* . This is all we need to know about M,.
r(
3. The T1 Theorem
415
Inspired by Lemma 1.4, we put M,P,f = (P,j')M,l + (M,Pt-f- (P,f)M,l) and rewrite (3.4) as
say. To estimate Al observe that, as in Proposition 1.3, it is enough to show that
where c is a constant independent of$ Let then g E C r ( R " ) ,llg1I2S 1, and bound
= A3 *
Ad, say. That A4 s c follows at once from the Littlewood-Paley theory. To bound A3 we invoke Theorem 1.1 in Chapter XV and note that it does not exceed cII T1 II*IIN(P,f)II,s cII T1 11*llf12. This implies that the limit defining A, exists and that its L2 norm is less than of equal to cllfl12, which is an estimate of the right order. Finally to bound A2, once again we let g E C;(R"), llg1I2 S 1, and observe that
say. As before, ASS c. To estimate A6 it clearly suffices to bound each of the integrals
XVZ. Cauchy Integrals on Lipschitz Curves
416
But this is not hard; indeed, from Holder's inequality and (3.5) it readily follows that for each j the above expression does not exceed
Furthermore, since
r
we get that if 0 < a < S',
This gives immediately that the limit defining A2 exists and that its L2 norm ccllfl12, which is precisely what we wanted to show.
4. NOTES; FURTHER RESULTS AND PROBLEMS The topics discussed in this chapter have their origin in the theory of linear partial differential equations. As Calder6n [19781 explains it, the question is one of constructing an algebra, under composition, of differential, or more generally, pseudo-differential, operators. The problem of proving that the composition of two such operators is another operator of the same kind can be reduced to the following problem: let M adenote, in the one variable case, the operator multiplication by the Lipschitz function a, and show that HM, is an operator of the same type. Since HMO = MaH + [El,Ma], it is sufficient to show that [H, M a ] D is bounded in Lp, 1 < p < a.This Calder6n did in 1965 with the aid of the theory of analytic functions. The idea goes as follows: without great difficulty the problem H D ] is bounded in Lp, and this operator can reduces to showing that [Ma, be represented by
4. Notes; Further Results and Problems
417
that is, the so-called first commutator. This integral, as well as that representing the higher order commutators, are special cases of
where F is analytic in a neighborhood of IzI =ssup(la(x) - a ( y ) l / l x - y l ) . Several classical integrals, including the Cauchy integral along a curve r, are also special cases of the integral. After a change of variables, and with z(A) = x + iAa(x) and w(A) = y + iAa(y), we are reduced to consider
For A = 0 this operator coincides with H. Also differentiating with respect to A, we get the operator
whose analogy with Calder6n's first commutator is clear. Calder6n [ 1977al succeeded in using the ,mehods of the first commutator together with a weighted L2 estimate for the Lusin function and obtained (d/dA)llA(A)llS IIB(A)ll s c ( l + IIA(A))))2, where the norms are the operator norms in L2 and c is a constant which depends on the Lipschitz constant of a. From this differential inequality, and the fact that IIA(0)ll equals the norm of the Hilbert transform H, it follows that llA(1)ll c < a,provided that Ila'll.. S M, some finite constant. David [1982], [1984] removed this unnecessary restiction on M by means of a bootstrap argument. The proof given here, though, is a real variable one and is due to Coifman, McIntosh, and Meyer [1982a]. It is based on some ideas of Coifman and Meyer [1975], [1978], who settled in 1975 the case of the second commutator and soon afterwards extended their results to commutators of arbitrary order; the results of C. P. Calder6n [1975], [1979] are relevant here. The proof of Proposition 1.6 is from the work of Coifman, Meyer, and Stein [ 19833 and that of Theorem 3.1 is from the work of David and JournC [1984] and Coifman and Meyer [ 19851. Further Results and Problems Assume k ( x , y ) verifies the assumptions of 8.12 in Chapter XI, and k(x, y ) f ( y )dy denote the Calder6n-Zygmund operator let K f ( x ) = p.v. associated with it. Furthermore, let a E BMO( R " ) and consider the commutator T f ( x ) = [ M a ,K ] f ( x ) of multiplication by a and K. Show that T is 4.1
I,.
418
XVI. Cauchy Integrals on Lipschitz Curves
bounded in L P ( R n ) ,1 < p < oo, with 'norm s clla ll*. (Hint: The desired conclusion follows immediately from the pointwise estimate ( ~ f ) # ( xG) c l l a l l * ( ( M ( I ~ f l ' ) ( x ) "+~M ( [flS)(x)'/",1 < r, s < a. Fix a cube I, then, and note that T f ( x ) = [Ma-,,, K ]f ( x ) = ( a ( x ) - a I ) K f ( x )- K ( ( a a , ) f ~ ~ ~) (Kx( )( a - a I ) f ~ R n \ 2 1 ) (= x )A + B + C, say. To estimate the average of I A ( over I observe that
The average of IBI over I
provided that qu = s. To bound C, let x, = center of I and note that for x in I,
4. Notes; Further Results and Problems
419
The result is from Coifman, Rochberg, and Weiss [1976], the above proof is due to Stromberg. In fact the following converse also holds: if [Ma, Rj] is bounded in some LP(Rn), 1 < p < co, and 1 s j s n, then a is in BMO(Rn) and Ilall* s c Cj(norm in LP of [Ma, Rj]). Uchiyama [I9781 has shown that T is compact from LP(Rn)into itself, 1 < p < a , if and only if a is in the BMO(Rn) closure of C:(Rn).) 4.2 The following extension of the results of 4.1 is due to Janson [1978]: Let 1 < p < co, and let 4 and be nondecreasing, positive functions on [O,W) connected by the relation 4 ( t ) = tn/q+-l(t-n), or equivalently I+-'(t) = t1'P4(t-'/n). We assume that I+ is convex, $(O) = 0 and +(2t) s cI+(t). If K is a homogeneous Calder6n-Zygmund operator, then a belongs to BMO,(Rn) if and only if [Ma, K ] maps LP(Rn)boundedly into L,(Rn). 4 3 Let
+
denote the Riesz potential of order a of f; and consider Tf(x) = [Ma, I,] f(x), the commutator of multiplication by the BMO(Rn) function a and I,. Show that T maps LP(Rn)into Lq(Rn),where l / q = I/p - a / n and 1 < p < n l a . (Hint: The proof follows along the lines of 4.1. The estimate for the A term involves the maximal function M, introduced in (2.16) in Chapter VI, but is otherwise similar to the A term in 4.1. The bound for the B term requires some care with the indices and uses the fact that the Riesz potentials I, map LP(Rn)into Lq(Rn), l / q = l / p - a l n . Finally it is readily seen that the C term is less than or equal to
The result is due to Chanillo [1982], who also discusses a converse.) 4.4 The following fact about operators is used repeatedly in Section 1 (cf. Proposition 1.3 for instance): suppose T,, Z,, and St are bounded, linear operators on a Hilbert space H, depending continuously (in the strong topology) on t. Suppose IISth 112 dt)
IlS(-,ll = SUP(! C0.m)
< a,
XVI. Cauchy Integrals on Lipschitz Curves
420
where the sup is taken over h E H with llhll = 1, and similarly for 11 T,.,II. Then j[,,,, s T z , ~ , d trepresents a bounded operator on H with norm ~sup,,,llZ,ll IlS,.,ll 11 T,.,II. Furthermore, if Z, = I and S, = T,, then Iljro,m, STS, dt 11 = Ilsc.,l12.This property, as well as an interesting discussion of the "Hilbert space methods" required for the proof of Calder6n's theorem, is in the work of Coifman, McIntosh, and Meyer [1982b]. 4.5 Let m E L m ( R ) be an even function and consider the variant of the Hilbert transform defined by Hm = p.v. R , m ( t ) / tdt. The techniques of Section 1 can be used to show that Hm is bounded in L 2 ( R ) .The kernel k,(x - y ) of Hm is obtained from the odd function k m ( x )whose restriction to ( 0 , ~is)k , ( x ) = e - x u m ( l / u )du. Given n complex valued, Lipschitz functions, A , , . . . ,A,, say, let k,,,(x) be a singular integral kernel defined by n ! k , , , ( x , y ) = ( A , ( x )- A I ( Y ) ) . . - A n ( y ) ) D n k m (-x Y ) . Show that the norm in L 2 ( R ) of the operator with kernel k,,,(x, y ) is less than or equal to c ( l + n)411m11m11~;llm. IIA'n((OT. A variant of, actually a corollary to, this result states that a similar conclusion holds for the principal value operator with kernel
I,
where k is now the odd function whose restriction to ( 0 , W ) is given by e - x u m ( u y )du. An immediate consequence of this observation is the following: let K be a compact, convex subset of the complex plan, F an analytic function on a neighborhood of K, and A a complex-valued, Lipschitz function such that if x # y, ( A ( x )- A ( y ) ) / ( x- y ) E K . Then for an odd kernel k as above, the singular kernel F ( ( A ( x )- A ( y ) ) / ( xy ) ) k ( x - y ) defines a bounded operator in L 2 ( ~ These ). results are from the work of Coifman, McIntosh, and Meyer [1982a]. 4.6 Coifman and Meyer [I9781 developed a method of dealing with commutators by reducing them to certain multilinear operators. In the bilinear case, where th: t$chniques they use are already apparent, the result reads as follows: Let 4, be C F ( R n )functions so that at least one of them vanishes in a neighborhood of the origin, and let
jlo.m,
+
where m ( t ) is a bounded function. Then
+.
6
6
where c depends only on 4 and (Hint: Suppose first both and vanish near the origin, and let 7; be a compactly supported, even function,
42 1
4. Notes; Further Results and Problems
which is 1 in a neighborhood of the support of h E C,"(Rn),
6 and
$.
Then for
R"
say. Since a E B M O ( R n ) ,la * ~ , h , ( x ) l ~ ( ldxdt / r ) is a Carleson measure and A cllall * llh1I2. Also B s 11 fl12, and we have finished in this case. The general result follows readily from this; the above proof is from Calderon dxdt is a [1978]. Since by 4.17 in Chapter XV also la * cCr,(x)12w(x)(l/r) Carleson measure for w in A,, with constant independent in A,, the reader is invited to consider the general weighted version of (4.1).) 4.7 There is a weighted version of Theorem 1.10. Corresponding to the operator in (1.22), let
and C*(+, f ) ( x ) = sup,lC,(A f ) ( x ) l . Then there are constants, k , , k2 such that for all 1 < p < the following inequality holds:
)~2 where c is independent of f, d p ( x ) = ( M ( ( 1+ ~ $ ( 4 ' ) ) ~ l ) ( xand M r g ( x ) = s u p ( ( l / l I l )S,lg(x) - g,Iq dx)'Iq, where the sup is taken over all open cubes containing x. This result is due to Krikeles [1983]. 4.8 These are some examples of operators T which verify the weak boundedness property, ( 1 ) Let k be a standard C Z kernel so that k ( x , y ) = -k(-y, x ) , x # y in Rn. Then T f ( g ) = lim,,, ~ l x - y l , , k ( x , y ) f ( y ) g ( x )dy dx defines an operator from Y ( R n )into Y ' ( R n ) with the weak boundedness property. Indeed, using the antisymmetry of the kernel we have
Tf ( g ) = lim -
422
XVI. Cauchy Integrals on Lipschitz Curues
Thus the smoothness o f f and g compensates for the singularity of k and the limit is easily seen to exist and to verify the desired properties. (2) Suppose $ is a smooth function with 0 integral and let Q,g denote the operator convolution with $,, i.e., Q,g = g * $,. If
where the L,'s are uniformly bounded operators on L'(R"), then T has the weak boundedness property. Indeed, T is well defined from Y ( R n )into Y ' ( R n ) by T f ( g ) = J I o . , , ( ~ & Q,g) ( l l t ) dt in the first case and by J r o . m , ( ~L&T ~ () l l t ) dt in the second case. These examples are from the work of David and JournC [1984]. 4.9 Suppose k is a CZ kernel which verifies the assumptions of 8.12 in Chapter XI with index 0 < S S 1 and let Tf denote the p.v. operator associated to k If T*l = 0, then T extends to a continuous operator from H P ( R n )into itself, n / ( n + 6) < p s 1. (Hint: By 6.6 in Chapter XIV it sufficesto show that T maps atoms into appropriate molecules. Let then a be a ( p , q, N) atom, supp a G B, where B is a ball centered at x and radius r; we show that Ta is a ( p , q, a ) molecule based at B(x, 2r). Since T is ~ ' ~the '. bounded in L q ( R n )we have that 11 Ta 11; S clla 11; S ~ l a 1 ~ r ~ " -On other hand
and
say. To estimate each A, observe that
this estimate replaced in each A, allows us to sum that expression and to
4. Notes; Further Results and Problems
423
conclude that it is of the right order as well. Finally, it only remains to check that property (iii) of molecules holds, namely, Ta(y) dy = 0. But observe that 0 = T * l ( a ) = Ta(1) = Ta(y) dy. This result, which is due to Alvarez and Milman [1985], can be extended in several directions. The above mentioned authors consider, for instance, operators analogous to those intoduced in 8.17 in Chapter XII.)
5,.
I,.
CHAPTER
XVII Boundary Value Problems on C’-Domains
1. THE DOUBLE AND SINGLE LAYER POTENTIALS ON A C’-DOMAIN
We say that a tempered distribution T is a fundamental solution for the Laplacian A in R” if AT = 6, the Dirac delta at 0. Fundamental solutions are useful for u = T * f solves A u = f: It is not hard to findAT explicitly. Suppose first n > 2 and note that since AT = 6, also - 151’T(5) = 1, and consequently, at least formally,
General considerations show that T (x) coincides with a function homogeneous of degree - ( n - 2) in R”\(O), but since the constants involved are important we compute them. For tMs purpose recall that
which substituted into (1.1) gives T ( x )= -
I
[o,m)
t
1
IRn
e-rlcle’x.t & d t =
(2~)”
-
I
tP(x, t ) dt,
[0,4
where P ( x , t) denotes the Poisson kernel of R:+’. Whence by the results in Chapter X,
424
1. Double,and Single Layer Potentials on a c ' - ~ o m a i n %
425
where wn is the surface area of the unit ball in Rn.Also T ( x )= ( 1 1 2 ~lnlxl, ) n = 2. We now turn to the boundary value problems. We restrict ourselves to C' domains D in Rn, the precise definition will be given below, and denote points there with capital letters, X, Y, . . . if they lie in the interior of D and P, Q if they lie on the boundary dD of D. Let also dQ denote the surface area element on aD. If u is a harmonic function in a bounded domain D, differentiable up to the boundary, then Green's identity (4.5) in Chapter VII and a limiting argument give
where (d/aNQ)denotes the derivative along the inward normal NQ into D. A quick verification of (1.2) goes as follows: Since for X, Y in D, u ( Y ) A T ( X - Y ) - T ( X - Y ) A u ( Y )= u ( Y ) S ( X - Y ) , the left-hand side of Green's identity is u( Y ) 6 ( X - Y ) d Y = u ( X ) ,and the right-hand side is that of (1.2), as claimed. Identity (1.2) is the starting point for the solution of the Dirichlet and Neumann problems in a bounded domain D. The method we use here, called that of double- and single-layer potentials, involves properties of integral equations on the boundary aD, covered by the Fredholm theory. The first term in (1.2) is the single-layer potential of ( a / a N Q ) yand since the singularity of T is of degree lower than the dimension of the boundary this potential is continuous on D and its closure. The second term in (1.2) is the double-layer potential of u and since the singularity of the kernel (d/aNQ)Tis of order n - 1, some care is needed as this singularity is only integrable if the domain is C1+", E > 0. As in 6.10 in Chapter VII the Dirichlet problem on D is stated as follows: Given f E LP(aD),find a function u so that Au = 0 in D and ulaD= J: To solve this problem we form the double-layer potential v off and observe that if f and aD are smooth enough, the boundary values of v equal (fI+ K ) f , where K is a compact linear mapping when D is C'. Methods from the Fredholm theory of integral operators yield the invertibility of f I + K, and the harmonic function u given by the double-layer potential of (;I + K ) - ' f then solves the Dirichlet problem. To do the Neumann problem, i.e., to find a function u so that Au = 0 in D and (a/aNQ)uldD = g, we use the single layer potential instead. To make all these statements precise we need some preliminary results. Throughout, D denotes an open, connected, bounded subset of R n such that R"\D is also connected.
ID
426
XVII. Boundary Value Problems on C1-~omains
Definition 1.1. We say that D is a C1 domain, and by this we really mean that D is a smooth n-manifold so that its boundary a D is a C1(n - 1)-manifold without boundary, if the following properties hold: to each Q in a D there corresponds a local coordinate system ( UQ, 4Q)such that
(i) UQ is an open neighborhood of Q and 4Qis a real-valued, compactly supported C'(Rn-') function defined on UQ. (ii) In the local Euclidean coordinates we may assume that Q = (0,4(0)). (iii) 4(O) = 0 and ( a / a ~ ~ ) 4 ( x )= l ~0.= ~ (iv) D n UQ = { ( x , t ) : x ~Rn-l, t ~ R and 4 ( x ) < t ) n UQ. (v) a D n UQ = {(x, t): x E R"-', t E R and +(x) = t ) n UQ. Remark 1.2. On account of the compactness of a D and property (iii) above it readily follows that given E > 0 we can find a finite number of coordinate systems, { ( q , 4j))z1 say, so that a D G U;=, q, (i)-(v) above hold, and IIV4,Illm E, all j. Remark 1.3. If 4 is any of the 4 Q ' above, ~ it is clear that there exists a sequence {4j) of C,"(Rn-') functions so that on the support of 4, (i) +j converges uniformly to 4, as j + oo, (ii) V+j converges uniformly to V4, as j + m. Moreover, since 4 is compactly supported, the +j's may be explicitly constructed as a sequence of mollifiers of 4, and consequently also IIV+jlII, S c, all j. The first step is to make use of the local coordinates to find the Euclidean expression of the layer potentials and to study them, as well as the traces on the boundary a D itself, as operators on LP. First observe that since
the double-layer potential off is given by
As for the boundary, or trace, double-layer potential it is defined as follows: for P E a D let
1. Double and Single Layer Potentials on a C'-Domain
427
and, when it makes sense,
Similarly, the single-layer potential off is given by
and since the singularity of ( P - Q12-" is integrable, the corresponding trace is
To obtain the Euclidean expression of the operator in (1.4) we localize as follows: Let { V,}be a finite cover of aD obtained as in Remark 1.2, say, and let { q m }be a nonnegative, smooth, finite partition of unity subordinate to the y's. Clearly,
and it suffices to consider each summand separately. Also by collecting all summands that correspond to each q.we may assume that we are working , we denote simply by with a fixed local coordinate system (V,, c $ ~ )which (U,9).Next if we change in U the variables Q into (y, + ( y ) ) , and put P = (x, 4(x)) and 4!lE = {y E R"-': Ix - y12 + (4(x) - 4 ( y ) ) ' > s2},since NQdQ = (-V4(y), 1) dy, the resulting expression in (1.4) equals (1/wn) k(x, Y)(C,, vrn,(y, ~ ( Y ) ) M~Y( ,Y I d ) ~where ,
I,,
Note that if 4 E C2(R"-'),then Ik(x, y)l zs cIx - yl-n+2 and the kernel is locally integrable; in this case most of the complicated arguments we give for the case when 4 is merely C' are unnecessary. At any rate, and with an obvious abuse of notation, it is clear that the Lp continuity properties of the operator in (1.4) follow from those of the Euclidean operator (1.9)
As a first approximation to study (1.9) we consider (1.10)
428
XVII. Boundary Value Problems on C1-Domains
We then have Theorem 1.4. Assume 4 is a Lipschitz function in R"-' such that IIV411m = r) < CO. Let k be the kernel in (1.8) and kJ the operator defined in (1.10). Then the mapping k * f ( x ) = ~up~,~IkJ(x)Iis bounded in LP(R"-'),1 < p < 00, with norm which goes to 0 with r). Furthermore k&x) = k f ( x ) exists pointwise a.e. and in Lp, and k is compact in LP(R"-').
Prmf. Write
k(x7Y ,
=
(Ix - yI2 n
-
4 ( x ) - 44Y) (+(x) - 4(y))2)n'2 (alayj ) 4 ( Y ) (xj - yj)
+
j=1
say, and put k i f ( x ) = ~lx-y,,B k,(x, y ) f ( y ) dy, 0 s j s n. Thus
and consequently, with a self-explanatory notation, (1.11)
That each of the k'*'sis bounded in LP(R"-'),1 < p < 00, follows immediately from Proposition 2.5 in Chapter XVI. When j = 0 we apply that result with A(x) = B ( x ) = # ( x ) , and in the remaining cases we put A ( x ) = + ( x ) and B(x) = xj, 1 S j d n, respectively. We also get that the norm c,, of ko* goes to 0 with llVBllm = IlVc$llrn = 77 and that the norm of the remaining depends only on r). Whence from (1.11) it readily follows that
zi*'s
(co + ndcllfllp, and the continuity assertion concerning I?* holds. Next, to prove that lim.,o k J ( x ) exists pointwise a.e. and in L p ( R " - ' ) , consider the sequence {4j}introduced in Remark 1.3 above, let (1.12)
and put kj,ef(X) =Jlx--yl,E &(x, y)f(y) dy. As before, it is readily seen that k ? f ( x > = SU~,,~~K~,J(X)I is bounded in Lp, 1 < p < 00; we claim that also
1. Double and Single Layer Potentials on a C'-Domain
429
limEdokj,,f(x) exists pointwise a.e. for f in LP(Rn-').Indeed, since each kj verifies Ikj(x, y)l s CIX - Y ( - " + ~where , c depends on j , we see at once that 1,.-11x - yl-n+21f(y)ldy < co a.e., and our claim is a simple consequence of this. Suppose now that f is real valued and let (1.13) L(X) = lim sup kEf(x)- lim inf XJ(X). 6-0
E+O
We want to show that L(x) = 0 a.e. First note that Rf(x)=
J;
(k(x, Y ) - &(x,Y ) ) ~ ( Ydy ) + kj,zf(x)
X-yI>€
=~
,J(x+ ) kj,.J(x),
(1.14)
say, where the kernel of Hj,, is given by an expression similar to (1.12) but is bounded in with C#J~ replaced by 4 - 4j there. Thus ~up,>~~H~,,f(x)I LP(Rn-'),1 < p < 00, with norm cj which goes to 0 with IlV(4 - 4j)llm. This is all we need to know; indeed, from (1.13) and (1.14) we get L(x) = lim sup q,,f(x) - lim inf q , f ( x ) E+O
E+O
s 2 sup l&.€f(X)l. E>O
Therefore, for each A > 0 and j, A P J { L > A}! S AP({sup,IHj,,I > A/2}( S 2P~jP(lf((g + 0 as j + co. Thus [ { L> A}l = 0 for each A > 0, L(x) = 0 a.e., and lim,+o k,f(x) = kf(x) exists a.e. That the convergence is also in LP(Rn-'),1 < p < co,follows at once from this last result, the boundedness of k* in LP(Rn-'),and the Lebesgue domjnated convergence theorem. Finally, we show that for each 1 < p < co, K is compact on LP(R"-').It is well known, and readily verified, that it is sufficient to exhibit a sequence of compact operators which convtrge to k in norm. As observed in the preceding paragraph limj+ml[k- Kill = 0, where 11 TI1 denote? the norm of T as a mapping on LP(Rn-').Next we show that also lim,+ollKj - kj,,II = 0. By a partition of unity argument we may restrict our attention to L P ( B ) , B = unit ball of Rn-'. In this case, and with a constant c that depends on j,
XVII. Boundary Value Problems on C'-Domains
430
where l / p + l/p' = 1. Thus, jBlkjf(x) - kj,&f(x)IPdx =s c ~ ~ ' ~ ' ~ ~ l f (dy, y)(~ and Ilkj - Zj,&IId C E " ~ ' + 0 with E. To check that each kj,& is compact, we must show that given a bounded sequence {f,} in L P ( B ) ,i.e., llfrnllP s M, all m, there exists a subsequence {f,,} such that ~bounded as a function of y and consequently is in LP'(B),it readily follows that kj,&f(x)= lim,,,, kj,&fmk(x). Moreover, since also ~ l k j , ElIkj,&fmk f ~ ~ p1, , cM, where c depends on E but is otherwise independent of the functions involved, by the Lebesgue dominated convergence theorem we obtain that 11 Kj,& fKj,& fmk 1, + 0 as mk + 00, and we are done. We turn now to the study of the trace of the double-layer potential.
Theorem 1.5. Let K , f ( P ) be the truncated trace double layer potential corresponding to a C' domain given by (1.4). Then the mapping K * f ( P) = sup,>,lK,f(P)I is bounded in LP(dD),1 < p < 00, limE+oK , f ( P ) = Kf(P) exists pointwise a.e. and in LP(dD),and K is compact in L P ( d D ) . Proof. By means of a partition of unity argument and by passing to local coordinates, the Lp boundedness of K * f ( P ) is readily seen to follow from the corresponding statement for the Euclidean operator K*f(x) = s ~ p , > ~ I K ~ f ( xwhere ) I , KEf(x)is defined in (1.9). We begin by showing that SUP)KJ(X)- kJ(x)l
S
cM~(x),
x
E
R"-',
(1.15)
&SO
where kJ(x) is the operator in (1.10) and c is an absolute constant which depends only on the Lipschitz constant 77 of 4. Since we work in local coordinates we may assume that x = 0, +(x) = 4(0)= 0, and V+(O) = 0; also the fact that 4 is C' means that I4(y)I = o(ly)) and )V4(y)l = o(ly1) as IyI + 0. Let then %& = {Iy12+ 4 ( y ) 2> E ~ } and observe that since B(0, ~ / ( 1 +q2)"*) E R"\%& c B(0, E ) we have x%,(y) = ~ ' V I , ( Y ) X B ( O , E ) ( Y+ ) XR"\B(O,&)(Y)* Thus
&f(O)
=
+
jRn-,W ,
Y ) ~ ~ . ( Y ) ~ ~ c o , & dy , ( Y ~ ~(1.16) (Y)
and consequently
y)l Furthermore, since (k(0,
d
271yl'-", the expression on the right-hand
431
1 . Double and Single Layer Potentials on a C'-Domain
side of (1.17) does not exceed 2r](l+ . 1 2 ) ' " - 1 ) % p A I E>O
E
If(y)I d y s CMf(O), IYlSE
and (1.15) follows. In turn, (1.15) gives K * f ( x ) S k * f ( x ) + cMf(x) , and consequently by Theorem 1.4 IIK*fII, s cllfll,, 1 < p < a,as we wanted to show. To prove the existence of the p.v. integral K f ( P ) assume first that f E C'(dD) and observe that
say. Since IAl s c jaDIP - QIz-"dQ < 00, the limit of this term is readily seen to exist as E + 0. As for B, let D E ( P )= {X E D :IX - PI > E } and note - Q ) dQ = 0. Whence that by 6.11 in Chapter VII, jdDe(P)(d/dNQ)T(P
and this last integral is readily seen to tend to 5 as E + 0 at each point P where the plane tangent to D is well defined. This proves the everywhere existence of the p.v. integral Kf(P ) when f is smooth. That this p.v. integral exists a.e. and in Lp(dD)for,an arbitrary f in Lp(dD),1 < p < 00, follows by a by now familiar argument which is left to the reader. Finally, we show that for each 1 < p < 00, K is compact on LP(dD);the assumption that D is a C' domain is needed here. Again through the use of a partition of unity argument, and on account of Theorem 1.4, our conclusion will follow from the qualitative version of (1.15), namely, K ~ ( x= ) @(x)
a.e.
(1.18)
In fact, (1.18) holds for those x's for which either side, and consequently the other side also, is well defined. Suppose x = 0 is such a point and observe that it suffices to show that the integral in (1.16) goes to 0 with E. But this is not hard; indeed, since 4 E C', Jk(0,y)l o ( ~ Y ~ ) / as ~ YI Y ~~ "+ 0, and the integral does not exceed
s co(l)Mf(O)
=
o(1) as
E
+ 0.
XVII. Boundary Value Problems on C'-Domains
432
Next we consider the behavior of the double layer potential Kf(X)given by (1.3) for X near the boundary d D of D. Since the notion of nontangential convergence is appropriate here we begin by defining cones interior to D. Cones in R" with vertex at 0 are given by {x = (x,, . . . ,x,):Ixl< fixn, p > l}, and this definition reads in our setting as follows: given 0 a < 1 and P E aD, the (inner) cone T,(P) with vertex at P and opening a is
-=
T,(P)
={X E
D : JX- PJ< S
and
a J X- PI < X - P . N p } . (1.19)
The constant 6 in the definition depends on a and D but is independent of P, and N p denotes as usual the inward normal at P. Similarly, the outer cone Tz(P) with the vertex at P and opening a is defined as
I'E(P) = { X E R"\D: IX - PI < S
and
alX - PI < - ( X - P ) . N p } . ( 1.20)
Given 0 < a < 1, P E dD and a function u ( X ) in D, we say that the nontangential limit (of order a ) of u ( X ) as X approaches P is L provided u ( X ) = L. Also the nontangential maximal function that limx-,p,xE~~(p) N,u(P)is N,U(P)= sup (U(X)I. (1.21) xcr,(p)
We then have Theorem 1.6. Let K f ( X ) be the double layer potential corresponding to a
1. Double and Single Layer Potentials on a C'-Domain
433
C' domain D given by (1.3). Then iff E LP(dD),1 < p < 00, and 0 < a < 1, there is a number S which depends only on a and D, so that for this choice of 6 in definition (1.19), ( 1.22)
where c depends only on p and 6. Furthermore K f ( X )converges nontangentially of order a a.e. on aD, and limx+RxGra(p)K f ( X ) = g(P)+ K f ( P ) , a.e. on dD, where K f ( P ) is the trace double layer potential in (1.5).
u ,;
Proof. Let dD E Bj, where each of the balls Bj = B ( < , aj) corre~ ~ ~to ) l V 4 ~ ( sponds to a coordinate system (Bj, c,bj) so that ~ ~ p ~ ( ~ ~S, A/6; obtain this covering apply Remark 1.2. Now let S = min(6, ,. . . ,6,) be the value in the definition (1.19) of the cones r , ( P ) . For P in dD we want to estimate N,(Kf)(P).By a partition of unity argument we may assume that suppf c Bj, some j. We consider two cases, to wit, (i) P E B(4,36,.) (nearby points) and (ii) P & B ( 4 , 3 ~ 3(far ~ ) away points). Case (ii) is easily handled. Since we are interested in estimating K f ( X ) for X in T , ( P ) and suppf E Bj, we must bound the integral in (1.3) when ( X - PI s 6,lpj - QI s 6j and ( P - $1 z-3Sj. Then also IX - QI Sj and
In other words Na(Kf)(p)xBCP,~S,)(p)
cllfllp-
(1.23)
Case (i) requires some work. First note that, if x E r , ( P ) , then IX - pl.1 s 4Sj, and consequently passing to the local coordinates given by + j 7 which we denote by #J from now on, we have IIVc$llm s a/6. Thus identifying Q = (y, 4 ( y ) ) , X = (x, t ) and P = (xo, +(x0)), the consideration of N,(Kf)(P)reduces to the study of tr I ( 1.24) where (1.25)
subject to
XVII. Boundary Value Problems on C1-Domains
434 and
(1.27) is readily seen to imply
+ To estimate (1.24) we break up the integral there in two parts, IJl y -l d sMI J~ly-xol,MJ = I J, say where M = max(3)x- x,l, t - +(xo)).To estimate I, observe that by (1.28) M t - +(xo) and lk(x, t, y)l S c(lt - O(y)l + I X - yl)'-" in the integral. Since as is readily seen (1.28) also implies that ( 4 / 5 ) ( t- +(xo))s It - c#~(y)l+ (a/6)ly - X I , we immediately get
+
-
If(Y)I dy s cMf(x0).
I s c(t - +(x,))l-"
(1.29)
Next we estimate
say. Clearly, 53 = ~
i M (x0)I f
i * f( ~ 0 ) .
As for J, note that Ik(x, t, y ) - k(xo, t, y)l S clx - xol/ly - xoln, whenever J y- x,J > 31x - xol. This estimate follows easily from the mean value theorem. Thus by Proposition 2.3 in Chapter IV, Jl s c
I
Ix - xol If(Y)I dY s cMf(xo). l ~ - x o l ~ 3 l x - I~ Y l - xol
(1.31)
Similarly, since Ik(xo, t, y ) - k(x0, +(x0),Y ) I s c ( t - ~ ( x o ) ) / ~ Yxol" whenever ly - xol > t - +(x,), we also have J2 s cMf (x,).
(1.32)
43 5
1. Double and Single Layer Potentials on a C 1 - ~ o m a i n
Whence adding estimates c(K*f(xo)+ ~ f ( x o ) )and ,
11 Tf [ I p
d
(1.29)-(1.32)
cIIK*fIIp + IIMfIIp
cIIf IIp,
we
get
that
1
0, or g E Lq(dD), 1 < q < 00 if l / p S 1 / ( n - 1). In either case, since the norm of +K'+ is small on Lq,we conclude that .If,and consequently f itself, belongs to Lq(dD),p < q. Iterating this process we get that f E Lq(aD),1 < q < 00, as anticipated. Let now u ( X ) denote the single layer potential of the function f over dD given by (1.7), and consider the integral I = jRm,DIVu(X)12 dX.
2. The Dirichlet and Neumann Problems
441
If a'D denotes the boundary of R"\D (it coincides with aD except for the orientation), then by Green's theorem div(Vu(X)) dX
a
=
u ( Q ) dQ,
(2.2)
where (a/aNb)indicates the derivative in the direction of the inward normal N', = -NO into a'D. The application of Green's theorem is justified since by Theorem 1.9 the last integral in (2.2) is absolutely convergent. Also by Theorem 1.9 ( a / a N b ) u ( Q )= - ($1+ K ' ) f ( Q ) = 0, Q a.e. in aD, and consequently 1 = 0. Therefore u(X) is constant on R"\D, and since limlxkmu(X) = 0 and R"\d is connected, then u(X) is identically 0 in R"\D. Furthermore, since u(X) is a continuous function on R" and u ( ,= ~ 0, by the uniqueness principle of harmonic functions, Proposition 4.3 in Chapter VII, we obtain that u(X) is identically 0 on R". From Theorem 1.9 it now follows that also (fZ K ' ) f ( Q )= 0 a.e. on aD, and consequently f(Q) = ($1 K 'If( Q ) + (fZ - K 'If( Q) = 0 a.e. on aD. In other words, fZ + K' is injective, and the proof is complete.
+
+
+
Corollary 2.5. $1 K is invertible on L:(aD), 1 < p < m. We are now ready to prove the existence and uniqueness of the solution to the Dirichlet problem.
Theorem 2.6. Suppose D is a C' domain and R"\D is connected. Given f E Lp(aD), 1 < p < m, there exists a unique harmonic function u(X) defined for X in D, such that for each 0 < a < 1, there exists a S > 0 which depends only on a and D, so that for this choice of 6 in definition (1.19), N,u belongs to Lp ( d o ) and IlNa~Ilp
(2.3)
~llfllp,
with c independent of$ Moreover, limx+.p,xcra(P)u(X)
= f(P)a.e.
on aD.
Proof. By Theorem 2.4 and Proposition 2.3, ;I + K has a continuous inverse in Lp(aD),1 < p < 00. Let u(X) be the double-layer potential u(X) = ( l / W n ) IjD((X - 0. N Q ) I ( I X- Ql"))(fZ + K ) - ' f ( Q ) dQ. BY Theorem 1-6 the nontangential limit of u(X) is f(P) a.e. on aD, and ~~IV,UI~~ S which is (2.3). cII(fZ + K)-'fll, s The proof of the uniqueness requires some work. For X, Y , in D and Q E aD, let F ( X , Q ) = (41 K)-'( I/lX - * I"-')( Q ) and consider the Green's function G(X, Y)defined by
+
XVII. Boundary Value Problems on C -Domains
442
Next, for fixed E > 0 consider the set D, = { Y E D : dist( Y , dD) s E } and s CEI"~, let $&(Y ) E C r ( D )satisfy0 s +bE s 1, = 1 on 0, and la"/dY"$&,,l where c depends only on a.For a fixed X in D, and for small E, by Green's identity we see that $E
U(x)=
U ( X ) $ k ( X )=
ID
G ( X , y ) A(u&k) ( y )d y
(2.4)
Moreover, if u is harmonic in D, integrating by parts (2.4) gives u(X) = -2
[
-
-
V&(X, Y ) Vt,be(Y ) u (Y ) d Y
JD
I,a x ,
Y ) A$&
Y )d Y
=A+B, say. We will show that under the additional assumption that the nontangential boundary values of u are 0, then A, B + 0 with E, and consequently u vanishes identically. Since the proofs for A and B are similar we only do A here. For this purpose let { $ j } be a finite family of nonnegative, C r ( R " ) functions, such that C $j( Y ) = 1 on { Y E R":dist( Y , d o ) S 6 ) and supp qj c B,, where (B,, 4,) is a local coordinate system for D. It clearly suffices to show that for each j , lj = j D l v y G ( X ,Y)IIV$&(Y)Ilu(Y)l$,(Y) dY, goes to 0 with E. Fixj, put 4 = I, $- = $, 4j = 4, and passing the variable Y to Euclidean coordinates note that I
S E
I I
l0,&lV&(X, Y, t + 4(Y))IlU(Y, t + 4(Y))l dtdY
lYl==C
b ( Y , t + 4J(Y))l dtdy. SUP IVYG(X, Y , s + 4J(Y))l e c lylee O S S S & E lo,&) Since G ( X , 0 )E Ll(aD) for each 1 < Q < 00, supoSsSelV&(X, y , s + 4 ( y ) ) l C N,(IV&()(y, 4 ( y ) ) E L 4 ( { y :lyl < c}). It is also easy to see that there is 0 < a < 1, so that ~ u p ~ . = ~ & (ty+, +(y))l d N,u(y, ~ ( J J ) )There. fore, if in addition to being harmonic, u verifies N,u(y, 4 ( y ) )E L p ( { y :Iyl e c}) and u(y, t + 4 J ( y ) )+. 0 as t + 0 for almost every lyl =sc, then IAl+ 0 with E, and the proof is complete.
Concerning the regularity properties of the solution to the Dirichlet problem we have Theorem 2.7. Suppose D is as in Theorem 2.6. Iff E Lf(dD),1 < p < 00, then the solution u ( X ) of the Dirichlet problem given by Theorem 2.6 has
2. The Dirichlet and Neumann Problems
443
the additional property that N,(lVul) E Lp(dD)and there is a constant c, independent off, so that
l l ~ ~ ( l V ~ l ) l lcpl l f l l ~ : ( a ~ ~ The proof of this result, being analogous to that of Theorem 2.6 is omitted; Theorem 1.7 is relevant here. Finally, we consider the Neumann problem; we begin by showing
Theorem 2.8. Suppose D is a bounded, connected, C' domain, and let K ' be the trace single layer potential defined by (1.35). Then for each 1 < p < 00, fI - K ' is invertible on the subspace of LP(dD)consisting of those functions f with I,, f(Q ) dQ = 0.
Prmf. Since by Theorem 1.8, K ' is compact, by Proposition 2.3 it is enough to prove that { I - K ' is injective. So assume that f = 2KLf and j , , f ( Q ) dQ = 0. As in Theorem 2.4 we conclude thatf E Lq(dD)for every 1 < q < 00. Let now u ( X ) denote the single layer potential o f f over dD defined by (1.7). Integrating by parts we get
Hence u ( X ) is constant in D. In R"\D, u ( X ) is harmonic and u ( X ) = 0. As noted ulaD= c, a constant. Since the maximum or limlxl+m minimum of u in R"\D are assumed on dD, then they both occur at every P in dD and the nontangential limit of ( d / d N , ) u ( X ) as X + P, X E R"\D is of constant sign. But by Theorem 1.9 the limit in our case is - g ( P ) KLf(P) = -f(P).Thus f is of constant sign, and since it has vanishing integral we must have f(P)= 0 on dD. W We are now ready to prove the existence and uniqueness of the solution to the Neumann problem.
Theorem 2.9. Suppose D is a bounded, connected, C' domain and R"\D is connected. Given g E Lp(dD),1 < p < 00, with I,, g ( Q ) dQ = 0, there exists a unique harmonic function u ( X ) defined for X in D such that for each 0 < a < 1, we can find a S > 0 which depends only on a and D, so that forthis choice of 6 in definition (1.19), Na(IVul)belongs to Lp(dD)and
IINal
