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00,
if and only if
(1.7)
Proof. The necessity of (1.7) follows from the uniform boundedness principle.
To see that it is sufficient, first note that if 9 is a trigonometric polynomial, then SN9 = 9 for N > degg. Therefore, since the trigonometric polynomials are dense in LP (see Corollary 1.11), if f E LP we can find a trigonometric polynomial g such that Ilf  gllp < E, and so for N sufficiently large
o If 1 < p < 00, then inequality (1.7) holds, as we will show in Chapter 3. When p = 1, the L1 operator norm of S N is again L N, and so by Lemma 1.7 the answer to the first question is no. When p = 2, the functions {e27fikx} form an orthonormal system (by (1.3)) which is complete (i.e. an orthonormal basis) by the density of the trigonometric polynomials in L2. Therefore, we can apply the theory of Hilbert spaces to get the following.
5. Summability methods
9
Theorem 1.9. The mapping f ~ {j(k)} is an isometry from L2 to £2, that ~s,
= 2::
Ilfll§ =
Ij(k)12.
k== Convergence in norm in L2 follows from this immediately. The second question is much more difficult. A. Kolmogorov (1926) gave an example of an integrable function whose Fourier series diverges at every point, so the answer is no if p = 1. If f E LP, 1 < p < 00, then the Fourier series of f converges almost everywhere. This was shown by L. Carleson (1965, p = 2) and R. Hunt (1967, p > 1). Until the result by Carleson, the answer was unknown even for f continuous.
5. Summability methods In order to recover a function f from its Fourier coefficients it would be convenient to find some other method than taking the limit of the partial sums of its Fourier series since, as we have seen, this approach does not always work well. One such method, Cesaro summability, consists in taking the limit of the arithmetic means of the partial sums. As is well known, if lim ak exists then .
a1
hm k= also exists and has the same value.
+ ... + ak k
Define 1 aN f(x)
=
N
N
+
=1
1
2:: Skf(x) k=O
1
1
f(t) N
o
=
l
+
N
1
2:: Dk(X  t) dt k=O
!(t)FN(X  t) dt,
where FN is the Fejer kernel,
FN(t) =
1 N
tDk(t) =
+ 1 k=O
1 N
+
FN has the following properties: FN(t) > 0,
(sin(1f.(N + 1)t))2 1 sm(1ft)
10
1. Fourier Series and Integrals
(1.8) lim
IIFNIlI =
[FN(t) dt
f
FN (t) dt = 0 if 0 > O.
N?= J8
= 1,
Because FN is positive, its L1 norm coincides with its integral and is 1. This is not the case for the Dirichlet kernel: its integral equals 1 because of cancellation between its positive and negative parts while its L1 norm tends to infinity with N.
Theorem 1.10. If f E V, 1 then
Proof. Since
00,
or if f is continuous and p
=
00,
J FN = 1, by Minkowski's inequality we have that
IlaN f  flip = 11/2 IIfC  t)  f(·) IIpFN(t) dt 1/2
<
f
J tl<8
IIfC  t)  f(')lIpFN(t) dt + 211fllp
1
Since for 1
f
FN(t) dt.
J8
< 00,
ETa IIf(' 
t)  f(·) lip = 0,
and the same limit holds if p = 00 and f is continuous, the first term can be made as small as desired by choosing a suitable O. And for fixed 0, by (1.8) the second term tends to O. 0
Corollary 1.11.
(1) The trigonometric polynomials are dense in LP, 1
00.
(2) If f is integrable and j(k) = 0 for all k, then f is identically zero. A second summability method is gotten by treating a Fourier series as the formal limit on the unit circle (in the complex plane) of
(1.9)
u(z) =
=
L
1
L
j(k)zk +
k=O
j(k)zlkl,
z = re 27riO .
k==
Since {j(k)} is a bounded sequence, this function is well defined on Izl It can be rewritten as
u(re 27riO ) =
=
L
k==
j(k) r k e27rikO = 1
1
1
1/2
1/2
f(t)Pr (()

t) dt,
< 1.
6. The Fourier transform of L1 functions
11
where
is the Poisson kernel. The Poisson kernel has properties analogous to those of the Fejer kernel:
l
(1.10)
Pr(t) > 0, Pr(t) dt = 1,
lim ( Pr (t) dt r.1 J8
a result
Theorem 1.12. If f E LP, 1 then
= 0 if 6" > O.
analogous to Theorem 1.10.
lim IIPr r.1
*f
00/\
or if f is continuous and p
 flip
Since the function u is harmonic on Iz I Dirichlet problem: ~u
=0 u=f
=
00,
= O. <
1, it is the solution to the
if Iz I < 1, if Iz I = 1,
where the boundary condition is interpreted in terms of Theorem 1.12. In Chapter 2 we will study the almost everywhere convergence of (J N f (x) and Pr * f(x).
6. The Fourier transform of L1 functions Given a function
f
E L1(JRn ), define its Fourier transform by
j(~) = I f(x)e27rix.~ dx, . )jRn
(1.11)
where x . ~ = x16 + x26 of the Fourier transform: (1.12) (1.13) (1.14) (1.15)
+ ... + xn~n.
The following is a list of properties
+ /3g)"'= oJ + /3fJ (linearity); Iljlloo < Ilflh and j is continuous;
(af
lim j(~)
I~I'oo
= 0 (RiemannLebesgue);
(f * g)"'= jfJ;
1. Fourier Series and Integrals
12
(1.16)
(Thfr(~)
=
1(~)e21rih.f;" where Thf(x)
(fe21rih.xr(~)
(1.17)
=
I(~
= f(x + h);
 h);
if pEOn (an orthogonal transformation), then
(f(p'))l~) = I(p~);
= Anf(>..lx), then g(~) =
(1.18)
if g(x)
(1.19)
.( 8 f ) ~ 8xj (~) = 27ri~jf(~);
(1.20)
(2"ixjfn/J =
I(A~);
;~ (~)
The continuity of 1 follows from the dominated convergence theorem; (1.14) can be proved like Lemma 1.4; the rest follow from a change of variables, Fubini's theorem and integration by parts. In (1.19) we assume that 8f j8xj E LI and in (1.20) that xjf ELI. Unlike on the torus, LI(JRn ) does not contain LP(JRn ) , P > 1, so (1.11) does not define the Fourier transform of functions in those spaces. For the same reason, the formula which should allow us to recover f from
I,
I,
may not make sense since (1.13) and (1.14) are all that we know about and they do not imply that is integrable. (In fact, is generally not integrable. )
1
1
7. The Schwartz class and tempered distributions A function f is in the Schwartz class, S(JR n ), if it is infinitely differentiable and if all of its derivatives decrease rapidly at infinity; that is, if for all a.,/3 E Nn , sup IxC! D/J f(x)1
= Pa,/J(f) < 00.
x
Functions in ergo are in S, but so are functions like e 1x12 which do not have compact support. The collection {Pa,/J} is a countable family of seminorms on S, and we can use it to define a topology on S: a sequence {
7. The Schwartz class and tempered distributions
13
With this topology S is a Frechet space (complete and metrizable) and is dense in LP(IRn ), 1 < p < 00. In particular, SeLl and (1.11) defines the Fourier transform of a function in S. The space of bounded linear functionals on S, S', is called the space of tempered distributions. A linear map T from S to C is in S' if lim T(¢k)
k '> 00
=0
whenever
lim ¢k
k '> 00
=0
in S.
Theorem 1.13. The Fourier transform is a continuous map from S to S such that
r fff = JF.nr jg.J'Rn
(1.21 )
i
and (1.22)
Equality (1.22) is referred to as the inversion formula. To prove Theorem 1.13 we need to compute the Fourier transform of a particular function.
Lemma 1.14. If f(x)
= errlxl2 then
j(~) = errl~12.
Proof. We could prove this result directly by integrating in C, but we will give a different proof here. It is enough to prove this in one dimension, since in IRn j is the product of n identical integrals. The function f(x) = e rrx2 is the solution of the differential equation
u'
+ 27rxU = 0, u(O) = 1.
By (1.19) and (1.20) we see that with the initial value
U(O)
=
Therefore, by uniqueness,
J.
u satisfies the same differential equation
u(x) dx
=
fa
e rrx2 dx
=
1.
j = f.
Proof of Theorem 1.13. By (1.19) and (1.20) we have ~Cl D(3 j(~) = C(D Cl x(3 fr(~),
so
o
1. Fourier Series and Integrals
14
The Ll norm can be bounded by a finite linear combination of semi norms of f, which implies that the Fourier transform is a continuous map from S to itself. Equality (1.21) is an immediate consequence of Fubini's theorem since f(x)g(y) is integrable on IR n x IRn. From (1.18) and (1.21) we get
J f(x)g(Ax) dx = J j(x)Ang(A1x) dx. If we make the change of variables Ax
= y in the first integral, this becomes
J f(A1x)g(x) dx = J j(x)g(Alx) dx; if we then take the limit as A ~
00,
we get
f(O) J g(x) dx = g(O) J j(x) dx. Let g(x)
= e?r 1x I2 ; then by Lemma 1.14, f(O) =
which is (1.22) for x
f(x)
=
= O.
Jj(~) d~,
If we replace f by Txf, then by (1.16),
(Txf)(O) =
J(Txfr(~) d~ = J j(~)e2?riX{, d~.
o If we let
j (x) = f (x), we get the following corollary.
Corollary 1.15. For f E S, (jr = j, and so the Fourier transform has period 4 (i. e. if we apply it four times, we get the identity operator). Definition 1.16. The Fourier transform of T E S' is the tempered distribution T given by A
T(f)
= T(j),
f
ES.
By Theorem 1.13, T is a tempered distribution, and in particular, ifT is an integrable function, then T coincides with the Fourier transform defined by equation (1.11). Likewise, if p., is a finite Borel measure (i.e. a bounded linear functional on Co (IRn), the space of continuous functions which vanish at infinity), then fl is the bounded continuous function given by
fl(~) = ( e 2?rix.c; dp.,(x).
JRn
For 8, the Dirac measure at the origin, this gives us
J=
1.
8. The Fourier transform on LP, 1 < p < 2
15
Theorem 1.17. The Fourier transform is a bounded linear bijection from
S' to S' whose inverse is also bounded. Proof. If Tn
T in S', then for any f
t
E
S,
Furthermore, the Fourier transform has period 4, so its inverse is equivalent to applying it 3 times; therefore, its inverse is also continuous. 0 _ If we define
('rr = T.
T by T(f)
T(j), then it follows from Corollary 1.15 that
=
And if TELl then by the inversion formula we get that
T(x)
= ( T(e)e 27rix .t;, de; JJRn
in particular, T is a bounded, continuous function.
8. The Fourier transform on LP, 1 < p If f E £P, 1 < P < for ¢ E S define
00,
<2
f can be identified with a tempered distribution:
then
Clearly this integral is finite. To see that Tj is continuous, suppose that ¢k t 0 in S as k t 00. Then by Holder's inequality, ITj(¢k)1
< IlfllplI¢kllp'·
Then II¢kllp' is dominated by the L= norm of functions of the form Xa¢k, and so by a finite linear combination of seminorms of ¢k; hence, the lefthand side tends to 0 as k t 00. Moreover, when 1 < P < 2 we have that
f is a function.
Theorem 1.18. The Fourier transform is an isometry on L2; that is, i E L2 and IIil12 = Ilf112. Furthermore,
i(e)
= lim (
f(x)e 27rix .t;, dx
R+= J1xl
and f(x)
=
lim
{
i(e)e 27riX .t;, de,
R+= J1t;,I
where the limits are in L2. The identity
IIil12 = IIfl12 is referred to as the Plancherel theorem.
1. Fourier Series and Integrals
16 A
Proof. Given that
f, h
_
E 5, let g = h, so that fj = h. Then by (1.21) we have
(1.23) If we let h = f then we get IIfl12 = IIil12 for f E 5. Since 5 is dense in L2, the Fourier transform extends to all f in L2 with equality of norms.
Finally, the continuity of the Fourier transform implies the given formulas for f and i as limits in L2, since fXB(O,R) and iXB(O,R) converge to f and in L2. 0
i
If f E LP, 1 < P < 2, then it can be decomposed as f = h + 12, where hELl and 12 E L2. (For example, let h = fX{x:lf(x)l>l} and 12 = f  h.) Therefore, i = i1 + i2 E L oo + L2. However, by applying an interpolation theorem we can get a sharper result. Theorem 1.19 (RieszThorin Interpolation). Let 1 < PO,P1, go, q1 < and for 0 < () < 1 define P and q by
1  =
1()
()
1
+, 
=
1()
00,
()
+.
P Po PI q qo q1 If T is a linear operator from LPO + LPI to LqO + Lql such that
and
then
The proof of this result uses the socalled "threelines" theorem for analytic functions; it can be found, for example, in Stein and Weiss [18, Chapter 5] or Katznelson [10, Chapter 4]. Corollary 1. 20 (HausdorffYoung Inequality). If f E LP, 1 :S P < 2, then I f E LP and A
Proof. Apply Theorem 1.19 using inequality (1.13), Plancherel theorem, IIil12 = Ilf112.
Ililioo < Ilf111' and the 0
9. The convergence and summability of Fourier integrals
17
We digress to give another corollary of RieszThorin interpolation which is not directly related to the Fourier transform but which will be useful in later chapters. Corollary 1.21 (Young's Inequality). If f E LP and g E Lq, then f L T , where 11r + 1 = lip + 1/q, and
*g E
Ilf * gilT < Ilfllpllgllq· Proof. If we fix
f
E LP we immediately get the inequalities
and
The desired result follows by RieszThorin interpolation.
o
9. The convergence and summability of Fourier integrals The problem of recovering a function from its Fourier transform is similar to the same problem for Fourier series. We need to determine if and when lim R+oo
(
iBR
j(e)e 21riX .f, de = f(x),
where BR = {Rx : x E B}, B is an open convex neighborhood of the origin, and the limit is understood either as in LP or as pointwise almost everywhere. If we define the partial sum operator S R by
(SRff =
XBRj,
then this problem is equivalent to determining if lim SRf
R+oo
= j.
Analogous to Lemma 1.8, a necessary and sufficient condition for convergence in norm is that
where Cp is independent of R. When n = 1 this is the case; we will prove this in Chapter 3. We will also prove several partial results when n > 1, but in general there is no convergence in norm when p I 2. We will discuss this in Chapter 8. In the case n
= 1, if B = (1,1) then SRf(x) = DR * f(x),
18
1. Fourier Series and Integrals
where DR is the Dirichlet kernel,
rR e27rix~ d~ = sin(27fRx).
DR(X) =
J R
7fX
This is clearly not integrable, but it is in well defined if f E LP, 1 < p < 00.
Lq(~)
for any q > 1, so DR
*f
is
Almost everywhere convergence depends on the bound
II sup ISRflll p< Cpllfllp· R
This holds if 1 < p < it here.
00
(the CarlesonHunt theorem) but we cannot prove
For the Fourier transform, the method of Cesaro summability consists in taking integral averages of the partial sum operators,
oRf(x) =
~ foR Sd(x) dt,
and determining if limO"Rf(x) = f(x). When n = 1 and B = (1,1),
* f(x),
O"Rf(x) = FR where FR is the Fejer kernel,
(1.24)
R r FR(X) = R J Dt(x)
sin2(7fRx)
1
o
dt
= R(7fx)2 .
Unlike the Dirichlet kernel, the Fejer kernel is integrable. Since it has properties analogous to (1.8), one can prove that in LP, 1 < p < 00, lim O"Rf
R+oo
= f.
The proof is similar to that of Theorem 1.10. In: Chapter 2 we will prove two general results from which we can deduce convergence in LP and pointwise almost everywhere for this and the following summability methods. The method of AbelPoisson summability consists in introducing the factor e27rtl~1 into the inversion formula. Then for any t > 0 the integral converges, and we take the limit as t tends to o. If we instead introduce the factor e7rt21~12, we get the method of GaussWeierstrass summability. More precisely, we define the functions
(1.25) (1.26)
r e27rtl~lj(~)e27riX.~ d~, w(x, t) = r e7rt21~12 j(Oe27riX.~ d~, JRn u(x, t) =
JRn
and then try to determine if
(1.27)
lim u(x, t)
t+O+
=
f(x),
10. Notes and further results
19
lim w(x, t) = f(x)
(1.28)
t ........ O+
in LP or pointwise almost everywhere. One can show that u(x, t) is harmonic in the halfspace ]R~+1 = ]Rn x (0, (0). When n = 1 we have an equivalent formula analogous to (1.9):
(1.29)
u(z)
=
f=
io
j(~)e27rizf. d~ +
fO
i=
j(~)e27rizf. d~,
z
=
x + it,
which immediately implies that u is harmonic. The limit (1.27) can be interpreted as the boundary condition of the Dirichlet problem, on ]Rn+l + , x E ]Rn.
~u=O
u(x,O) = f(x), It follows from (1.25) that'
u(x, t) = Pt
* f(x),
where Pt(~) = e 27rt1 f.1. One can prove by a simple calclilationif n = 1, and a more difficult one when n > 1 (see Stein and Weiss [18, p. 6]), that
(130) .
.
Rt(x) =
r
(ntl )
t (2 1 12) n+l t+x 2
'~~ n+l 7r2
.
This is called the Poisson kernel. In the case of GaussWeierstrass summability, one can show that the function w(x, t) = w(x, J47rt) is the solution of the heat equation
ow

ot

_
~w =
0
w(x,O) = f(x)
on ]Rn+l + '
xE
]Rn,
and (1.28) can be interpreted as the initial condition for the problem. We also have the formula
w(x, t)
= Wt * f(x),
where Wt is the GaussWeierstrass kernel,
(1.31)
Wt(x) = tne7rlxI2 jt 2 •
This formula can be proved using Lemma 1.14 and (1.18).
10. Notes and further results 10.1. References.
The classic reference on trigonometric series is the book by Zygmund [21], which will also be a useful reference for results in the next few chapters. However, this work can be difficult, to consult at times. Another comprehensive reference on trigonometric series is the book by Bary [1].
1. Fourier Series and Integrals
20
There are excellent discussions of Fourier series and integrals in Katznelson [10] and Dym and McKean [4]. The book by R.E. Edwards [5] is an exhaustive study of Fourier series from a more modern perspective. The article by Weiss [20] and the book by Korner [12] are also recommended. An excellent historical account by J. P. Kahane on Fourier series and their influence on the development of mathematical concepts is found in the first half of [9]. The book Fourier Analysis and Boundary Value Problems, by E. GonzalezVelasco (Academic Press, New York, 1995), contains many applications of Fourier's method of separation of variables to partial differential equations and also contains historical information. (Also see by the same author, Connections in mathematical analysis: the case of Fourier series, Amer. Math. Monthly 99 (1992),427441.) The book by O. G. J0rsboe and L. Melbro (The CarlesonHunt Theorem on Fourier Series, Lecture Notes in Math. 911, SpringerVerlag, Berlin, 1982) is devoted to the proof of this theorem. The original references for this are the articles by L. Carleson (On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135157) and R. Hunt (On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp. 235255, Southern Illinois Univ. Press, Carbondale, 1968). Kolmogorov's example of an Ll function whose Fourier series diverges everywhere appeared in Une serie de FourierLebesgue divergente partout (C. R. Acad. Sci. Paris 183 (1926), 13271328). 10.2. Multiple Fourier series. Let ']fnbe the ndimensional torus (which we can identify with the quotient group ~n /'ZP). A function defined on ']fn is equivalent to a function on ~n which has period 1 in each variable. If fELl (~n) then we can define its Fourier coefficients by
j(v) =
J
f(x)e27rix.v dx,
and construct the Fourier series of
L
v E Zn ,
f with these coefficients, j(v)e27rix.v.
vE1P
One can prove several results similar to those for Fourier series in one variable, but one needs increasingly restrictive regularity conditions as n increases. See Stein and Weiss [18, Chapter 7]. 10.3. The Poisson summation formula. Let
f be a function such that for some <5 > 0, If(x)1 < A(l + Ix!)no
and
Ij(e)1 < A(l + lel)no.
21
10. Notes and further results
(In particular,
f
and
j
are both continuous.) Then
This equality (or more precisely, the case when x = 0) is known as the Poisson summation formula and is nothing more than the inversion formula. The lefthand side defines a function on ']fn whose Fourier coefficients are precisely j (v). 10.4. Gibbs phenomenon.
Let f(x) = sgn(x) on (1/2,1/2). By Dirichlet's criterion, for example, we know that S N f (x) converges to f (x) for all x. To the right of 0 the partial sums oscillate around 1 but, contrary to what one might expect, the amount by which they overstep 1 does not tend to 0 as N increases. One can show that
217r sin(y) dy
lim supSNf(x) = NHXJ
7r
X
0
Y
~
. 1.17898 .... .... _
This phenomenon occurs whenever a function has a jump discontinuity. It is named after J. Gibbs, who announced it in Nature 59 (1899), although it had already been discovered by H. Wilbraham in 1848. See Dym and McKean [4, Chapter 1] and the paper by E. Hewitt and R. E. Hewitt (The Gibbs Wilbraham phenomenon: an episode in Fourier analysis, Arch. Hist. Exact Sci. 21 (1979/80), 129160). Gibbs phenomenon is eliminated by replacing pointwise convergence by Cesaro summability. For if m < f (x) < M, then by the first two properties of Fejer kernels in (1.8), m < ON f(x) ::; M. In fact, it can be shown that if m < f(x) < M on an interval (a, b), then for any E > 0, m  E < ON f(x) < M + E on (a + E, b  E) for N sufficiently large. 10.5. The Hausdorff'Young inequality.
Corollary 1.20 was gotten by an immediate application of RieszThorin interpolation. But in fact a stronger inequality is true: if 1 < P < 2 then ~
1IJllpi <
(
p 1/P ) (p')I/pl
n/2
IIfll p·
This inequality is sharp since equality holds for f(x) = e7rlxI2. This result was proved by W. Beckner (Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159182); the special case when p is even was proved earlier by K. 1. Babenko (An inequality in the theory of Fourier integrals (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 531542).
1. Fourier Series and Integrals
22
In the same article, Beckner also proved a sharp version of Young's inequality (Corollary 1.21).
10.6. Eigenfunctions for the Fourier transform in L2(JR). Since the Fourier transform has period 4, if j is a function such that j = Aj, we must have that A4 = 1. Hence, A = ±1, ±i are the only possible eigenvalues of the Fourier transform. Lemma 1.14 shows that exp( _TrX2) is an eigenfunction associated with the eigenvalue 1. The Hermite functions give the remaining eigenfunctions: for n > 0, ( l)n dn hn(x) =  , eXP(Trx 2)d exp( _TrX2) n. xn satisfies hn = (i)nh n . If we normalize these functions,
en
n )n. 1n 2 ,]1/2h = IIhhn l1 2 [(4 Tr V ~n. n,
we get an orthonormal basis of L2(JR) such that
= n=O
Thus L2(JR) decomposes into the direct sum Ho E9 HI E9 H2 $ H 3 , where on the subspace H j , 0 < j < 3, the Fourier transform acts by multiplying functions by i j . This approach to defining the Fourier transform in L2(JR) is due to N. Wiener a.nd can be found in his book (The Fourier Integral and Certain of its Applications, original edition, 1933; Cambridge Univ. Press, Cambridge, 1988). Also see Dym and McKean [4, Chapter 2]. In higher dimensions, the eigenfunctions Qf the Fourier transform are products of Hermite functions, one in each coordinate variable. Also see Chapter 4, Section 7.2.
10.7. Interpolation of analytic families of operators. The RieszThorin interpolation theorem has a powerful generalization due to E. M. Stein. (See Stein and Weiss [18, Chapter 5].) Let S = {z E C : 0 < Re z < I} and let {Tz } zES be a family of operators. This family is said to be admissible if given two functions f, 9 E LI(JR n ), the mapping Zft
r Tz(j)gdx
JlRn
is analytic on the interior of S and continuous on the boundary, and if there exists a constant a < Tr such that eallmzllog is uniformly bounded for all
Z E
S.
r
JlR
n
Tz(f)g dx
10. Notes and further results
23
Theorem 1.22. Let {Tz } be an admissible family of operators, and suppose that for 1
IITiyfllqo < Mo(y)llfllpo where for some b <
and
II T l+iyfll q1 < Ml(y)llfll pl '
7f
supeblyllogMj(Y) <
00,
j
= 1,2.
yElR
e
e
Then for 0 < < 1, Re z = and p, q defined as in Theorem 1.19, there exists a constant Me such that
10.8. Fourier transforms of finite measures.
As we noted above, if P is a finite Borel measure then p, is a bounded, continuous function. The collection of all such functions obtained in this way is characterized by the following result. Theorem 1.23. If h is a bounded, continuous function, then the following are equivalent:
(1) h = P, for some positive, finite Borel measure Pi (2) h is positive definite: given any fELl (JR n ) ,
r r
JlRn JlRn
h(x  y)f(x)J(y) dxdy > O.
This theorem is due to S. Bochner (Lectures on Fourier Integrals, Princeton Univ. Press, Princeton, 1959; translated from Vorlesungen iiber Fouriersche Integrale, Akad. Verlag, Leipzig, 1932). Also see Katznelson [10, Chapter 6].
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
Chapter 2
The HardyLittlewood Maxilllal Function
1. Approximations of the identity Let ¢ be an integrable function on JRn such that J cP = 1, and for t > 0 define ¢t(x) = tn¢(tIx). As t ~ 0, ¢t converges in S' to 6, the Dirac measure at the origin: if 9 E S then
¢t(g)
=
r
J~n
tn¢(tIx)g(x) dx
r
=
J~n
cP(x)g(tx) dx,
and so by the dominated convergence theorem, lim cPt (g) = 9 (0) = 6 (g) .
ttO
Since 6 * 9 = g, for 9 E S we have the pointwise limit lim ¢t
ttO
* 9 (x) = 9 (x ) .
Because of this we say that {cPt: t
> O} is. an approximation of the identity.
The summability methods in the previous chapter can be thought of as approximations of the identity. For Cesaro summability, ¢ = FI and FR = ¢I/R (See (1.24).) For AbelPoisson summability, cP = PI (see (1.30)) and for GaussWeierstrass summability, ¢ = WI (see (1.31)). We see from the following result that these summability methods converge in £P norm. Theorem 2.1. Let {cPt: t
> O} be an approximation of the identity. Then lim II ¢t
ttO
if f E LP, 1
<
00,
*f
 flip
=
0
and uniformly (i.e. when p
= (0)
if f E Co(JR n ).

25
2. The HardyLittlewood Maximal Function
26
Proof. Because ¢ has integral 1, ¢t Given
E
* f(x)
 f(x) =
r ¢(y)[j(x  ty)  f(x)] dy.
JlR
n
> 0, choose 6 > 0 such that if Ihl < 6,
Ilf(' + h) 
E
f()llp <
211¢III'
(Note that 6 depends on f.) For fixed 6, if t is sufficiently small then
r
J1yl ?8/t 1¢(y)1 dy < 411;11 p . Therefore, by Minkowski's inequality
II¢t * f
 flip <
r
J1yl <8/t
1¢(y)lllf(' + ty) 
+ 211fllp
r
f(')llp dy
1¢(y)1 dy
J1yl ?8/t
< E. D
As a consequence of this theorem, we know that there exists a sequence { t k}, depending on f, such that t k  t 0 and lim ¢tk
k>oo
* f(x)
=
f(x)
a.e.
Hence, if lim.¢t * f(x) exists then it must equal f(x) almost everywhere. In Section 4 we will study the existence of this limit.
2. Weaktype inequalities and almost everywhere convergence Let (X,I£) and (Y, v) be measure spaces, and let T be an operator from LP(X,I£) into the space of measurable functions from Y to C. We say that T is weak (p, q), q < 00, if
v({y E Y: ITf(y)1 > A}) <
(GIl! lip )
q,
and we say that it is weak (p, (0) if it is a bounded operator from LP(X, 1£) to LOO(Y, v). We say that T is strong (p, q) if it is bounded from LP(X, 1£) to Lq(y, v). If T is strong (p, q) then it is weak (p, q): if we let E).. ITf(y)1 > A}, then
v(E ) =
).
r
JE)..
dv < 
r
JE)..
Tf(x) q dv < A 
IITfllg < Aq

= {y
(Cllfllp)q. A
E Y
2. Weaktype inequalities and almost everywhere convergence
27
When (X, JL) = (Y, v) and T is the identity, the weak (p,p) inequality is the classical Chebyshev inequality. The relationship between weak (p, q) inequalities and almost everywhere convergence is given by the following result. In it we assume that (X, JL) =
(Y, v). Theorem 2.2. Let {Ttl be a family of linear operators on LP(X, JL) and
define T* f(x) = sup ITtf(x)l· t
If T* is weak (p, q) then the set {f E LP(X, JL) : lim Ttf(x) = f(x) a.e.} t~to
is closed in LP(X, JL). T* is called the maxhnal operator associated with the family {Ttl. Proof. Let {fn} be a sequence of functions which converges to J in £P (X, JL) norm and such that Ttfn(x) converges to fn(x) almost everywhere. Then
JL({x EX: lim sup ITtf(x)  f(x)1 > \}) t~to
< JL({x EX: limsuplTt(f  fn)(x)  (f  fn)(x) I > \}) t~to
< JL( {x EX: T*(f  fn)(x) > \/2}) + JL({x EX: I(f  fn)(x) I > \/2}) 20 < ( Tllf  fnllp
)q + (2\ II!  fnllp )P ,
and the last term tends to 0 as n
7
00.
Therefore,
JL({x EX: lim sup ITtf(x)  f(x)1 > O}) t~to
00
< LJL({x EX: limsupITtf(x)  f(x)1 > l/k}) t~to
k=l
=0.
o By the same technique we can also prove that the set
{f E LP(X, JL) : lim Ttf(x) exists a.e.} t~to
is closed in LP(X, JL). It suffices to show that
JL ({ x EX: lim sup T t f (x)  lim inf T t f (x) > \}) = 0, t~to
t~to
2. The HardyLittlewood Maximal Function
28
and this follows as in the above argument since lim sup Ttf(x) liminfTtf(x) < 2T* f(x).
t7 to
t7 t o
(If Ttf(x) is complex, we apply this argument to its real and imaginary parts separately. ) Since for f E S approximations of the identity converge pointwise to f, we can apply this theorem to show pointwise convergence almost everywhere for f E LP, 1 < p < 00, or for f E Co if we can show that the maximal operator SUPt>o I(Pt * f (x) I is weakly bounded.
3. The Marcinkiewicz interpolation theorem Let (X, M) be a measure space and let f : X + C be a measurable function. We call the function aj : (0,00) + [0,00], given by
aj(>\) = fL({x EX: If(x)1 > A}), the distribution function of
f
(associated with fL).
Proposition 2.3. Let ¢ : [0,00) such that ¢(O) = 0. Then
Ix
+
¢(If(x)l) dfL =
[0,00) be differentiable, increasing and
10
00
¢'(A)aj(A) dA.
To prove this it is enough to observe that the lefthand side is equivalent to
Ix lI(X)'
¢i(A) dAdIJ.
and then change the order of integration. If, in particular, ¢(A)
= AP then
(2.1) Since weak inequalities measure the size of the distribution function, this representation of the LP norm is ideal for proving the following interpolation theorem, which will let us deduce LP. boundedness from weak inequalities. It applies to a larger class of operators than linear ones (note that maximal operators are not linear): an operator T from a vector space of measurable functions to measurable functions is sublinear if
IT(fo
+ fl)(x)1 <
ITfo(x) I + ITh(x)l,
IT(Af)1 = IAIITf!'
A E C.
3. The Marcinkiewicz interpolation theorem
29
Theorem 2.4 (Marcinkiewicz Interpolation). Let (X, p,) and (Y, v) be measure spaces, 1 < Po < PI < 00, and let T be a sublinear operator from LPO (X, p,) + LPI (X, p,) to the measurable functions on Y that is weak (Po, po) and weak (PI,PI). Then T is strong (p,p) for Po < P < Pl. Proof. Given f E LP, for each)"
= JI = fo
> 0 decompose f as fo + JI, where fX{x:lf(x)l>c).}, fX{x:lf(x)l:Sc).};
the constant c will be fixed below. Then fo E LPo (p,) and Furthermore,
JI
E LPI (p,) .
ITf(x)1 < ITfo(x) I + ITJI(x) I, so
We consider two cases.
Case 1: PI = 00. Choose c = 1/(2A I ), where Al is such that Alllgiloo. Then aTh ()../2) = o. By the weak (po,po) inequality, aT!o (>,/2)
<
C~o IlioIIPo )
IITglloo <
pO ;
hence,
r
IITfll~ < P roo )..pIPO(2Ao)p° io r
r1f(x)l/c
= p(2Ao)PO i x If(x) IPO io =
Case 2: PI <
If(x)IPO dp,d)"
i{x:lf(x)l>cA}
)..pIpo d)" dp,
P (2A o)PO(2A I )PPO IlfiI P. P Po P 00.
We now have the pair of inequalities i = 0,1.
From these we get (arguing as above) that
liTfll~ < P roo )..pIpo (2Ao)p° io
r
If(x) IPO dp, d)"
i{x:lf(x)l>c).}
+ P roo )..pIPI (2Ad PI io
r
If(x) IPI dp, d)"
i{x:lf(x)l:Sc).}
p2PI API) P2pO APO . = ( P  Po cP0po + PI  pCPPI I IlfiIP P
o
2. The HardyLittlewood Maximal Function
30
We can write the strong (p, p) norm inequality in this theorem more precisely as (2.2) where 1
()
1()
P
PI
Po
=+,
O<()
When PI = 00 this is the constant which appears in the proof; when PI < 00 it is enough to take c such that (2Aoc)PO = (2A I c)Pl and then simplify.
4. The HardyLittlewood maximal function Let Br = B(O, r) be the Euclidean ball of radius r centered at the origin. The HardyLittlewood maximal function of a locally integrable function f on IR. n is defined by (2.3)
M f(x) = sup IBI r>O
I
r
r
JBr
If(x  y) I dy.
(This can equal +00.) If we let ¢ = IBIIIXBl' then (2.3) coincides for nonnegative f with the maximal operator associated with the approximation of the identity {¢t} as in Theorem 2.2. Sometimes we will define the maximal function with cubes in place of balls. If Qr is the cube [r, r]n, define (2.4)
M' f(x) = sup ( l)n r>O
2r
r If(x  y) I dy.
JQr
When n = 1, M and M' coincide; if n > 1 then there exist constants en, depending only on n, such that
(2.5)
Cn
and
cnM'f(x) < Mf(x) < CnM'f(x).
Because of inequality (2.5), the two operators M and M' are essentially interchangeable, and we will use whichever is more appropriate, dep(;1nding on the circumstances. In fact, we can define a more general maximal function
(2.6)
M" f(x) =
~~~ I~I
k
If(y)1 dy,
where the supremum is taken over all cubes containing x. Again, Mil is pointwise equivalent to M. One sometimes distinguishes between M' and Mil by referring to the former as the centered and the latter as the noncentered maximal operator. Alternatively, we could define the noncentered maximal function with balls instead of cubes.
4. The HardyLittlewood maximal function
31
Theorem 2.5. The operator M is weak (1, 1) and strong (p,p), 1 < p
< 00.
We remark that by inequality (2.5), the same result is true for M' (and also for Mil). It is immediate from the definition that
(2.7) ,
11Mflloo < Ilflloo,
so by the Marcinkiewicz interpolation theorem, to prove Theorem 2.5 it will be enough to pr;ove that M is weak (1, 1). Here we will prove this when n = 1; we will prove the general case in Section 6. In the onedimensional case we need the following covering lemma whose simple proof we leave to the reader. Lemma 2.6. Let {Iah~EA be a collection of intervals in JR and let K be a compact set contained in their union. Then there exists a finite subcollection { I j } such that
c
K
UI
L Xlj (x) < 2,
and
j ,
j
x E JR.
j
Proof of Theorem 2.5 for n = 1. Let E).. = {x E JR : Mf(x) x E E).. then there exists an interval Ix centered at x such that
(2.8)
ILl
LIf I
> A}. If
>A
Let K c E),. be compact. Then K C U Ix, so by Lemma 2.6 there exists a finite collection {Ij} of intervals such that K C Uj I j and Lj Xlj < 2. Hence,
IKI < ~ IIjl < ~ ~ hJ. J
J
lfl <
h
~ ~Xljlfl < ~ Ilf111; J
the second inequality follows from (2.8). Since this inequality holds for every compact K C E),., the weak (1,1) inequality for M follows immediately. D Lemma 2.6 is not valid in dimensions greater than 1, and though one could replace it with similar results, this is not the approach we are going to take here. (For such a proof, see Section 8.6.) The importance of the maximal function in the study of approximations of the identity comes from the following result. Proposition 2.7. Let ¢ be a function which is positive, radial, decreasing (as a function on (0,00)) and integrable. Then
sup I¢t t>O
* f(x)1 < 11¢lh M f(x).
2. The HardyLittlewood Maximal Function
32
Proof. If we assume in addition to the given hypotheses that ¢ is a simple function, that is, it can be written as
¢(x)
=
L ajXBrj (x) j
with aj > 0, then 1
¢
* f(x) = ~ajlBrjllBrjlXBrj * f(x) <
' //¢//lMf(x)
J
since //¢//l = L ajlBrj I· An arbitrary function ¢ satisfying the hypotheses can be approximated by a sequence of simple functions which increase to it monotonically. Any dilation ¢t is another positive, radial, decreasing function with the same integral, and it will satisfy the same inequality. The desired conclusion follows at once. 0 Corollary 2.8. If 1¢(x)1 < 'lj;(x) almost everywhere, where'lj; is positive, radial, decreasing and integrable, then the maximal function SUPt I¢t * f (x) I is weak (1,1) and strong (p,p), 1 < p < 00.
This is an immediate consequence of Proposition 2.7 and Theorem 2.5. If we combine this corollary with Theorem 2.2 we get the following result. Corollary 2.9. Under the hypotheses of the previous corollary, if f E V, 1 < p < 00, or if f E Co, then
l~ qJ, * f(x) =
(J qJ) . f(x) a.e.
In particular, the summability methods discussed in Chapter 1, Section 9 (Cesaro, AbelPoisson and Gauss Weierstrass summability) , each converge to f(x) almost everywhere if f is in one of the given spaces.
Proof. Since we have convergence for f E 5,. by Theorem 2.2 we have convergence for f E 5 = LP (or f E Co if p = (0). The Poisson kernel (1.30) and the GaussWeierstrass kernel (1.31) are decreasing; the Fejer kernel (1.24) is not but Fl(X) < min(l, (7TX)2). 0
5. The dyadic maximal function In ]Rn we define the unit cube, open on the right, to be the set [0, 1)n, and we let Qo be the collection of cubes in]Rn which are congruent to [0, l)n and whose vertices lie on the lattice 7l n . If we dilate this family of cubes by a factor of 2 k we get the collection Qk, k E Z; that is, Qk is the family of
5. The dyadic maximal function
33
cubes, open on the right, whose vertices are adjacent points of the lattice (2 kz)n. The cubes in Uk Qk are called dyadic cubes. From this construction we immediately get the following properties:
(1) given x E ~n there is a unique cube in each family Qk which contains it;
(2) any two dyadic cubes are either disjoint or one is wholly contained in the other; (3) a dyadic cube in Qk is contained in a unique cube of each family Qj, j < k, and contains 2n dyadic cubes of Qk+1. Given a function
f
E Lfoc(~n),
Ekf(x)
=
define
L (,~,l f) XQ(x); QEQk
.
Q
Ekf is the conditional expectation of f with respect to the aalgebra generated by Qk. It satisfies the following fundamental identity: if 0 is the union of cubes in Qk, then
Ekf is a discrete analog of an approximation of the identity. The following theorem makes this precise; first, define the dyadic maximal function by (2.9)
Mdf(x)
=
sup IEkf(x)l· k
Theorem 2.10.
(1) The dyadic maximal function is weak (1, 1).
(2) If f E Lfoc! lim Ekf(x) k~=
= f(x)
a.e.
Proof. (1) Fix fELl; we may assume that f is nonnegative: if f is real, it can be decomposed into its positive and negative parts, and if it is complex, into its real and imaginary parts.
Now let
{x E ~n: Mdf(x) > A}
= UOk, k
where
Ok
= {x
E ~n :
Ekf(x) > A and Ejf(x) < A if j < k};
that is, x E Ok if Ekf(x) is the first conditional expectation of f which is greater than A. (Since f E L1, Ekf(x) ~ 0 as k ~ 00, so such a k exists.)
2. The HardyLittlewood Maximal Function
34
The sets Ok are clearly disjoint, and each one can be written as the union of cubes in Qk. Hence,
(2) This limit is clearly true if f is continuous, and so by Theorem 2.2 it holds for fELl. To complete the proof, note that if f E Ltoc then fXQ E L1 for any Q E Qo. Hence, (2) holds for almost every x E Q, and so for almost every x E }Rn. 0 This proof uses a decomposition of }Rn which has proved to be extremely useful. It is called the Calder6nZygmund decomposition and we state it precisely as follows. Theorem 2.11. Given a function f which is integrable and nonnegative, and given a positive number A, there exists a sequence {Qj} of disjoint dyadic cubes such that
(1)
f(x)
< A for almost every x
ret
UQj'j
(2) (3) Proof. As in the proof of Theorem 2.10, form the sets Ok and decompose each into disjoint dyadic cubes contained in Qk; together, all of these cubes form the family {Q j }.
Part (2) of the theorem is then just the weak (1,1) inequality of Theorem 2.10.
If x ret Uj Qj then for every k, Ekf(x) < A, and so by part (2) of Theorem 2.10, f(x) < A at almost every such point. Finally, by the definition of the sets Ok, the average of f over Q j is greater than A; this is the first inequality in (3). Furthermore, if Qj is the dyadic cube containing Qj whose sides are twice as long, then the average
6. The weak (1,1) inequality for the maximal function
of
35
f over Qj is at most .\.. Therefore, _1_
r
IQjl lQj
f < 10jl2:
r
f < 2n.\..
IQjllQjllQj
o 6. The weak (1,1) inequality for the maximal funct~n . We are now going to use Theorem 2.10 to prove Theorem 2.5. In fact, it is an immediate consequence of the following lemma and inequality (2.5). (Recall that M' is the maximal operator on cubes defined by (2.4).)
Lemma 2.12. If f is a nonnegative function, then
I{x E}Rn:
M'f(x) > 4n.\.} I < 2nl{x
E
jRn: Mdf(x) >
.\.}I.
Given this lemma, by the weak (1,1) inequality for Md proved in Theorem 2.10,
I{x E jRn : M'f(x) > .\.}I < 2n l{x E jRn : J'1I[df(x) > 4 n .\.} I < (Since M' f
= M'(lfl), we
may a~sume that
gn
~llflh
f is nonnegative.)
Proof of Lemma 2.12. As before, we form the decomposition
{x
E}Rn :
Mdf(x) > .\.}
=
UQj. j
Let 2Qj be the cube with the same center as Qj and whose sides are twice as long. To complete the proof it will suffice to show that
{x
E
jRn: M'f(x) > 4n.\.}
C
U2Qj. j
Fix x tf. Uj 2Qj and let Q be any cube centered at x. Let l(Q) denote the side length of Q, and choose k E Z such that 2k  1 < l(Q) < 2k. Then Q~ntersects m < 2n dy?dic cubes in Qk; call them Rl, R2,' .. ,Rm. None of these cubes is contained in any of the Qj's, for otherwise we would have x E Uj 2Qj. Hence, the average of f on each Ri is at most .\., and so
o
36
2. The HardyLittlewood Maximal Function
As a consequence of the weak (1,1) inequality and Theorem 2.2 we get a continuous analog of the second half of Theorem 2.10. Corollary 2.13 (Lebesgue Differentiation Theorem). If f E Ltoc(JRn ) then
lim IBI
r+O+
r
I
r
JBr f(x 
y) dy = f(x) a.e.
From this we see that If(x)1 < Mf(x) almost everywhere. The same is true if we replace M by M' or Mil. We can make the conclusion of Corollary 2.13 sharper: . hm
(2.10)
I1I
r+O+ Br
1 Br
If(x  y)  f(x)1 dy = 0 a.e.
This follows immediately from the fact that
I~rl
L"
If(x  y)  f(x)1 dy < Mf(x)
+ If(x)l·
The points in JRn for which limit (2.10) equals 0 are called theLebesgue poilltsof f. If x is a Lebesgue point and if {B j } is a sequence of balls such that B1 :=> B2 :=> .•. and j Bj = {x} (note that the balls need not be centered at x), then
n
lim _1_
r 'F= f(x).
IB j I JBj
j+oo
This follows immediately from the inclusion B j C B(x,2rj), where rj is the radius of B j . A similar argument shows that the set of Lebesgue points of f does not change if we take cubes instead of balls. The weak (1,1) inequality for M is a substitute for the strong (1,1) inequality, which is false. In fact, it never holds, as the following result shows. Proposition 2.14. If fELl and is not identically 0, then M f (j. L1.
The proof is simple: since that
Now if
Ixl > R,
BR
C
f
is not identically 0, there exists R
B(x, 2Ixl), so 1
Mf(x) >
r
(2Ixl)n JBR If I >
Nevertheless, we do have the following.
E
2n lxl n
'
> 0 such
7. A weighted norm inequality
37
ofrr~,n,
Theorem 2.15. If B is a bounded subset
then
where log + t = max (log t, 0). Proof.
Is
Mf <2
100
< 21BI Decompose
f
E
Mf(x) > 2\}1 d\
E B :
+ 2 1,00 I{x E B
h + 12, where h =
as
{x
I{x
B : Mf(x) > 2\}
: Mf(x)
fX{x:lf(x)I>A} C
{x
E
> 2\}1 d\.
and
12 =
f 
h.
Then
B : Mh(x) > \}.
Hence,
rOO I{x E B
: M f (x) > 2\} I d\ <
Jl
roo ~ ( Jl
1" lR.n
If (x) I dx d\ 
J{x:lf(x)I>A} I,max(lf(x)l,l) d\
If(x)1
\ 1
dx A
= C { If(x)llog+ If(x)1 dx. JlR.n
o 7. A weighted norm inequality Theorem 2.16. If w is a nonnegative, measurable function and 1 00,
then there exists a constant Cp such that
Furthermore, (2.11)
( J{x:Mf(x»A}
w(x) dx <
~l
(
If(x)IMw(x) dx.
JlR.n
Proof. It will suffice to show that 11M fllux)(w) < Ilfllu"'(Mw) and that the weak (1,1) inequality holds; the strong (p,p) inequality then follows from the Marcinkiewicz interpolation theorem. If Mw(x) = 0 for any x, then
2. The HardyLittlewood Maximal Function
38
w(x) = 0 almost everywhere and there is nothing to prove. Therefore, we may assume that for every x, Mw(x) > O. If a > IUlluXl(Mw) then f
1{x:lf(x)l>a}
Mw(x) dx = 0,
and so I{x E lRn : If(x)1 > a}1 = 0; that is, If(x)1 < a almost everywhere. From this it follows that M f(x) < a a.e., so 11M fIILOO(w) < a. Hence,
IIMfIlLoo(w) < IlfIILOO(Mw)· To prove the weak (1,1) inequality we may assume that f is nonnegative and f E LI(lRn). (If f E LI(Mw) then fj = fXB(O,j) is a sequence of integrable functions which increase pointwise to f.) If {Q j} is the Calder6nZygmund decomposition of f at height). > 0, then as we showed in the proof of Lemma 2.12,
{x
E
lR n : M'f(x) > 4n).}
U2Qj;
C
j
hence,
f
w( x) dx <
l{x:MI f(x»4n)..}
L j
=
f
w (x) dx
12Qj
L 2nlQjll 2Ql .1 j
J
f
w(x) dx
12Qj
<2;~ l f.f(y)( 12Ql.,1. J
< 2:C
Q)
L
J
W
(X)dx) dy
2Q)
j(y)M"w(y) dy.
Since M"w(y) < CnMw(y), we get the desired inequality.
o
If w is such that Mw(x) < w(x) a.e., then these inequalities hold with the same weight w on both sides. Functions which satisfy this condition are called Al weights, and we will consider them in greater detail in Chapter 7.
8. Notes and further results 8.1. References. The maximal function for n = 1 was introduced by G. H. Hardy and J. E. Littlewood (A maximal theorem with functiontheoretic applications, Acta Math. 54 (1930), 81116), and for n > 1 by N. Wiener (The ergodic theorem, Duke Math. J. 5 (1939), 118). In their article, Hardy and Littlewood first consider the discrete case, about which they say: "The problem
8. Notes and further results
39
is most easily grasped when stated in the language of cricket, or any other game in which a player compiles a series of scores of which an average is recorded." Their proof, which uses decreasing rearrangements (see 8.2 below) can be found in Zygmund [21]. Our proof follows the ideas of Calderon and Zygmund ( On the existence of certain singular integrals, Acta Math. 88 (1952), 85139). The decomposition which bears their name also first appeared in this paper. The method of rotations, discussed in Chapter 4, lets us deduce the strong (p, p) inequality for n > 1 from the onedimensional result. For a discussion of questions related to Theorem 2.2, and in particular the necessity of the condition in that theorem, see de Guzman [7]. The Marcinkiewicz interpolation theorem was announced by J. Marcinkiewicz (Sur l'interpolation d'operations, C. R. Acad. Sci. Paris 208 (1939), 12721273). However, he died in World War II and a complete proofwas finally given by A. Zygmund (On a theorem of M arcinkiewicz concerning interpolation of operations, J. Math. Pures Appl. 34 (1956), 223248). Both Marcinkiewicz interpolation and RieszThorin interpolation in Chapter 1 have been generalized considerably; see, for example the book by C. Bennett and R. Sharpley (Interpolation of Operators, Academic Press, New York, 1988). The weighted norm inequality for the maximal operator is due to C. Fefferman and E. M. Stein (Some maximal inequalities, Amer. J. Math. 93 (1971), 107115).
8.2. The Hardy operator, onesided maximal functions, and decreasing rearrangements of functions.
Given a function g on 1R+ defined by
Tg(t)
= (0,00)
the Hardy operator acting on g is
lint g(s) ds, t
=
t
E
1R+.
0
If 9 E L1 (1R+) is nonnegative, then, since Tg is continuous, one can show that
(2.12)
E(A) =
~r
/\ JE()..)
g(t) dt,
where E(A) = {t E lR+ : Tg(t) > A}. From this and (2.1) we get IITgllp < p'llgllp, 1 < p < 00. Other proofs of this result and generalizations of the operator can be found in the classical book by G. H. Hardy, J. E. Littlewood and G. Polya (Inequalities, Cambridge Univ. Press, Cambridge, 1987, first edition in 1932), or in Chapter 2 of the book by Bennett and Sharpley cited above.
40
2. The HardyLittlewood Maximal Function
For a function f on IR, the onesided HardyLittlewood maximal functions are defined by M+ f(t)
= sup h1it+h If(s)1 ds and h>O
M f(t)
= sup h1 i t If(s)1 ds. h>O
t
th
The maximal function as defined by Hardy and Littlewood corresponds to M for functions on IR+ (with 0 < h < t in the definition). When If I is decreasing this maximal function coincides with the Hardy operator acting on Ifl. If f is a measurable function on IR n , we can define a decreasing function f* on (0,00), called the decreasing rearrangement of f, that has the same distribution function as f:
f*(t) = inf{A : af(A) < t}. Because f and f* have the same distribution function, by (2.1) their LP norms are equal, as well as any other quantity which depends only on their distribution function. (Cf. Section 8.3 below.) The action of the Hardy operator on f* is usually denoted by f**. . Hardy and Littlewood showed that for functions on IR+, (2.13)
I{x : M f(x) > A}I < I{x : f**(x) > A}I,
so the weak (1,1) and strong (p,p) inequalities for M follow from the corresponding ones for T. A beautiful proof of the weak (1,1) inequality for M+ was given by F. Riesz as an application of his "rising sun lemma" (Sur un theoreme du maximum de MM. Hardy et Littlewood, J. London Math. Soc. 7 (1932), 1013). Given a function F on IR, a point x is a shadow point of F if there exists y > x such that F(y) > F(x). The set of shadow points of F is denoted by S (F) . Lemma 2.17. Let F be a continuous function such that
lim F(x)
X~+=
= 00 and
lim F(x)
x~=
=
+00.
Then S(F) is open and can be written as the disjoint union Uj(aj, bj ) of finite open intervals such that F (aj) = F (bj ) . Given f E Ll(IR) and A > 0, define F(x) = foX If(t)1 dt  AX. Then E(A) = {x : M+ f(x) > A} equals S(F). Using Lemma 2.17 one can deduce that
E(A)
=
~(
/\ JE(>..)
If(x) I dx,
8. Notes and further results
41
which is similar to (2.12) for the Hardy operator. The strong (p, p) inequality now follows as before with constant p' (which is sharp). We leave the details of the proof of this and of Lemma 2.17 to the reader. An inequality similar to (2.13) holds for the maximal operator acting on functions on ~n; in fact, we have the following pointwise inequality: there exist positive constants Cn and C n such that
cn(Mf)*(t) < f**(t) < Cn(Mf)*(t),
t
E ~+.
The lefthand inequality is due to F. Riesz; the righthand inequality is due to C. Herz (n = 1) and C. Bennet and R. Sharpley (n > 1). See Chapter 3 of the abovecited book by these authors for a proof and further references.
8.3. The Lorentz spaces
!y,q.
Let (X, JL) be a measure space. LP,q(X) denotes the space of measurable functions f which satisfy
Ilfllp,q = when 1 < p <
00,
1
(~1=[tl/Pf*(t)]q ~ty/q < 00,
00
and
Ilfllp,oo = sup t 1/ pf*(t) < 00 t>O
when 1
00.
When p
=
q,
Ilfllp,p = Ilf*llp = Ilfllp general, however, I . IIp,q
and we recover!Y. In is not a norm since the triangle inequality only holds when 1 < q < P < 00 or p = q = 00. But when 1 < p < 00 and 1 < q < 00, if we replace f* in the definition of Ilfllp,q with f**, we get a quantity which is equivalent to Ilfllp,q and which defines a norm. If ql < q2 then
11f11p,q2 < Ilfllp,qll so LP,ql
C
LP,q2.
An operator T is weak (p, p) precisely when
IITfllp,oo < Cllfll p· The Marcinkiewicz interpolation theorem can be generalized to these spaces; this lets us, for example, give a version of the HausdorffYoung inequality which is stronger than Corollary l.20: if f E LP(~n), 1 < p < 2, then I f E LP ,p and there exists a constant Bp such that A
Ilillp/,p < Bpllfllp· For more information on decreasing rearrangements and the LP,q spaces, see Stein and Weiss [18, Chapter 5] and the book by Bennett and Sharpley cited above. Also see this latter book for more on interpolation theorems, particularly the socalled real method of interpolation which is well suited
2. The HardyLittlewood Maximal Function
42
to Lorentz spaces. These spaces were introduced by G. G. Lorentz in two papers (Some new functional spaces, Ann. of Math. 51,(1950), 3755, and On the theory of spaces A, Pacific J. Math. 1 (1951),411429). 8.4. LlogL.
In the proof of Proposition 2.14, the integrability of M f failed at infinity, and did not exclude local integrability. However, the example f(x) = xI (log x )2 X(O,1/2] shows that even local integrability can fail. A partial converse of Theorem 2.15 is true and characterizes when M f is locally integrable. Theorem 2.18. If f is an integrable function supported on a compact set B, then Mf E LI(B) if and only if flog+ f E LI(B).
This is due to E. M. Stein (Note on the class L log L, Studia Math. 32 (1969), 305310); a proof can also be found in GarciaCuerva and Rubio de Francia [6, p. 146]. At the heart of the proof is a stronger version of the weak (1,1) inequality and a "reverse" weak (1,1) inequality for the maximal function:
(2.14) (2.15)
I{x
E
JRn : Mf(x)
> A}I <~ {
I{x E JRn : M f(x) > A}I >
J{x: If (x) 1>)../2}
~;;
If(x)1 dx,
If(x)1 dx.
{x:lf(x)I>)..}
Inequality (2.14) follows from the weak (1,1) inequality applied to h fX{x:lf(x)I>)"/2}; inequality (2.15) follows from the Calder6nZygmund decomposition if we replace M by Mil. In this and related problems it would be useful to consider functions f such that f log+ fELl as members of a Banach space. Unfortunately, the expression J f log+ f does not define a norm. One way around this is to introduce the Luxemburg norm: given a set 0 ~ JR n , define
(2.16)
IlfIiLlogL(l1) = inf
{A > 0: 10 If~)IIog+ Cf~)I) dx < I}.
By using this we can strengthen the conclusion of Theorem 2.18 to the following inequality: IIMfll£l(B) < CllfIiLlogL(B)' (See Zygmund [21, Chapter 4].) By replacing the function tlog+ t in (2.16) by any convex, increasing function <1>, we get a class of Banach function spaces, L
8. Notes and further results
43
Groningen, 1961), and M. M. Rao and Z. D. Ren, (Theory of Orlicz Spaces, Marcel Dekker, New York, 1991). 8.5. The size of constants.
In the proofs in this chapter, the constants which appear in the weak (1,1) and strong (p,p) inequalities, 1 < p < 00, for M and M' are of exponential type with exponent n. For the strong (p, p) inequality, the method of rotations (see Corollary 4.7) gives a better constant of the form Cpn for M. Furthermore, one can show that there exist constants Cp , independent of n, such that 1
00.
This result is due to E. M. Stein, and the proof appeared in an article by E. M. Stein and J.O. Stromberg (Behavior of maximal functions in]Rn for large n, Ark. Mat. 21 (1983), 259269); also see E. M. Stein, Three variations on the theme of maximal functions (Recent Progress in Fourier Analysis, 1. Peral and J. L. Rubio de Francia, eds., pp. 229244, NorthHolland, Amsterdam, 1985). A partial generalization of this result is possible: let B be a convex set which is symmetric about the origin and define MBf(x)
Then if p that
= sup I IBI1 r>O r
If(x  y)1 dy.
rB
> 3/2 there exists a constant Cp independent of Band
n such
For the weak (1,1) inequality, the best constant known for M is of order n. (See the article by Stein and Stromberg cited above.) For the noncentered maximal function defined by (2.6) (but with balls instead of cubes), the best constant in the strong (p, p) inequality must grow exponentially in n. See L. Grafakos and S. MontgomerySmith (Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1997), 6064). We note that when n = 1 the best constants are known for the onesided and for the noncentered maximal operator Mil but not for M. In the case of the weak (1,1) inequality for Mil, one can use Lemma 2.6 to get C = 2, and the function f(t) = X[o,l] shows this is the best possible. For the strong (p, p) inequality for Mil, see the article by Grafakos and MontgomerySmith cited above. For lower and upper bounds on the weak (1,1) constants for M, see the article by J. M. Aldaz (Remarks on the HardyLittlewood maximal function, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 19).
44
2. The HardyLittlewood Maximal Function
8.6. Covering lemmas.
A standard approach to proving that the maximal fmiction in JRn is weak (1,1) is to use covering lemmas. Here we give two; the first is a Vitalitype lemma due to N. Wiener (in the paper cited above). If B = B(x, r) and t > 0, then we let tB = B(x, tr). Theorem 2.19. Let {Bj hEJ be a collection of balls in JR n . Then there exists an at most countable subcollection of disjoint balls {B k } such that
U B C U5B j
jEJ
k.
k
The second is due independently to A. Besicovitch and A. P. Morse; for a proof and further references, see the book by M. de Guzman (Differentiation of Integrals in ]Rn, Lecture Notes in Math. 481, SpringerVerlag, Berlin, 1985). Theorem 2.20. Let A be a bounded set in JRn , and suppose that {Bx}xEA is a collection of balls such that Bx = B (x, r x), r x > 0. Then there exists an at most countable subcollection of balls {B j } and a constant Cn, depending only on the dimension, such that
A
C
UBj
and
LXBj(X) < Cn· j
J
The same result holds if balls are replaced by cubes; more generally, the point x need not be the center of the ball or cube but must be uniformly close to the center. Using the BesicovitchMorse lemma, we can extend our results for the maximal function to LP spaces with respect to other measures. Given a nonnegative Borel measure p" define the maximal function Mp,f(x)
= sup (~) r>O P,
r
r
} Br
If(x  y)1 dp,(y).
(If P,(Br) = 0, define the p,average of f on Br to be zero.) Then one can show that Mp, is weak (1,1) with respect to p,; hence, by interpolation it is bounded on LP(p,), 1 < p < 00. Note that if we define M~ as the maximal operator with respect to cubes, then Mp, and M~ need not be pointwise equivalent unless p, satisfies an additional doubling condition: there exists C such that for any ball B, p,(2B) < Cp,(B). Furthermore, if we define the noncentered maximal operator M~, then this need not be weak (1, 1) if n > 1 unless p, satisfies a doubling condition. In this case the weak (1,1) inequality follows from
8. Notes and further results
45
Wiener's lemma; when n = 1, Lemma 2.6 can be applied even to nondoubling measures. An example of JL such that M~ is not weak (1,1) is due to P. Sjogren (A remark on the maximal function for measures on JR n , Amer. J. Math. 105 (1983), 12311233). For certain kinds of measures a weaker hypothesis than doubling implies that M~ is weak (1,1); see, for example, the paper by A. Vargas (On the maximal function for rotation invariant measures in JRn , Studia Math. 110 (1994),917).
8.7. The nontangential Poisson maximal function. Given f E V(}Rn), u(x, t) = P t * f(x) defines the harmonic extension of f to the upper halfspace }R~:+I = }Rn x (0,00), and limt~O Pt * f(x) is the limit of this function on the boundary as we approach x "vertically", that is, along the line perpendicular to }Rn. More generally, one can consider a nontangential approach: fix a > 0 and let
ra(x) = {(y, t) : Iy 
xl < at}
be the cone with vertex x and aperture a. We can then ask whether lim
(y,t)~(x,O)
u(y, t) = f(x)
a.e. x E
}Rn.
(y,t)Er a(x)
Since this limit holds for all x if f is continuous and has compact support, it suffices to consider the maximal function u~(x)
=
sup
lu(y, t)l,
(y,t)Era(x)
so as to apply Theorem 2.2. But one can show that there exists a constant Ca , depending on a, such that u~(x)
and so
Ua
< CaM f(x), is weak (1,1) and strong (p,p), 1 < p < 00.
These results can be generalized to "tangential" approach regions, provided the associated maximal function is weakly bounded. For results in this direction, see, for example, the articles by A. Nagel and E. M. Stein (On certain maximal functions and approach regions, Adv. in Math. 54 (1984), 83106) and D. CruzUribe, C. J. Neugebauer and V. Olesen (Norm inequalities for the minimal and maximal operator, and differentiation of the integral, Publ. Mat. 41 (1997), 577604).
8.8. The strong maximal function. Let R(h I , .. . ,hn ) = [hI, hI] x ... x [hn, h n ]. If f E Lfoc' define the strong maximal function of f by
M,J(x) =
h,,~~Lo IR(hl,.l . , hn ) Il(h',
,hn ) If(x  y)1 dy.
2. The HardyLittlewood Maximal Function
46
Then Ms is bounded on LP(ffi.n) , p > 1; this is a consequence of the onedimensional result. However, Ms is not weak (1,1): the best possible inequality is
I{x E IRn
:
M,j(x) >
A}I < C
JIf~)1
(1 +log+
If~)I) n~l dx
This result is due to B. Jessen, J. Marcinkiewicz and A. Zygmund (Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217234). A geometric proof was given by A. Cordoba and R. Fefferman (A geometric proof of the strong maximal theorem, Ann. of Math. 102 (1975), 95100); also see the article by R. J. Bagby ( Maximal functions and rearrangements: some new proofs, Indiana Univ. Math. J. 32 (1983), 879891). The analog of the Lebesgue differentiation theorem, lim max(hi)'O
1 IR(h 1 , ... ,hn)1
r
f(x  y) dy = f(x) a.e.
JR(hl, ... ,hn )
is false for some fELl, but is true if f(l + log+ and in particular if f E Lfoc for some p > 1.
Ifl)n1
is locally integrable,
If in the definition of Ms we allow rectangles (that is, parallelepipeds) with arbitrary orientation (and not just with edges parallel to the coordinate axes), then the resulting operator is not bounded on any LP, p < 00, and the associated differentiation theorem does not hold even for f bounded.
For these and other problems related to differentiation of the integral see de Guzman [.7], the book by the same author cited above, and the monograph by A. M. Bruckner (Differentiation of integrals, Amer. Math. Monthly 78 (1971), Slaught Memorial Papers, 12). 8.9. The Kakeya maximal function.
Given N > 1, let RN be the set of all rectangles in ffi.n with n  1 sides of length h and one side of length N h, h > O. The Kakeya maximal function is defined by lCN f(x)
=
sup xERERN
IR11
rIf I·
JR
Since each rectangle in RN is contained in a ball of radius cnNh, the Kakeya maximal function is bounded pointwise by the HardyLittlewood maximal function:
(2.17) It follows immediately that lCN is bounded on LP, 1 < p <
00.
An important problem is to determine the size of the constants as functions of N for the LP estimates for the Kakeya maximal function. Inequality
8. Notes and further results
47
(2.17) gives the trivial estimate N n  1 for the constant in the weak (1,1) in"equality, and it is clear that in L=, lCN has norm 1. Interpolating between these values we get
IllCN flip < CnN(nl)/Pllfll p,
1
< p < 00.
It is conjectured that the constant can be improved to a power of log N when p = n, which, again by interpolation, would give the following bounds:
(1) if p > n then IllCNfllp < Cn(logN)apllfllp; (2) if 1 < p < n then IllCNfllp < CnNn/pl(logNYl!pllfll p· This conjecture is closely related to two other problems: the Hausdorff dimension of the Kakeya seta set with Lebesgue measure zero which contains a line segment in each direction (see Stein [17, p. 434]); and the boundedness properties of BochnerRiesz multipliers (see Chapter 8, Sections 5 and 8.3). This conjecture has been proved completely only when n = 2: see the paper by A. Cordoba (The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), 122). In this paper he also discussed the connection with BochnerRiesz multipliers. Cordoba later proved the result in all dimensions when 1 < p < 2; see A note on BochnerRiesz operators (Duke Math. J. 46 (1979), 505511). M. Christ, J. Duoandikoetxea and J. L. Rubio de Francia (Maximal operators related to the Radon transform and the Calder6nZygmund method of rotations, Duke Math. J. 53 (1986), 189209) extended the range to 1 < p < (n + 1)/2. A great deal of activity on this problem has been spurred by the work of J. Bourgain (B esicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), 147187). In this paper he proved the conjecture for 1 < p < (n + 1)/2 + En, where En is given by an inductive formula (for instance, E3 = 1/3). T. Wolff (An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), 651674) improved this to 1 < p < (n+2)/2. Recently, J. Bourgain (On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), 256282) has shown that there exists c > 1/2 such that the conjecture is true for p < en if n is large enough. For a discussion of recent results on this problem and its connection with the Kakeya set, see the survey article by T. Wolff (Recent work connected with the Kakeya problem, Prospects in Mathematics, H. Rossi, ed., pp. 129162, Amer. Math. Soc., Providence, 1999).
Chapter 3
The Hilbert Transform
1. The conjugate Poisson kernel Given a function f in S(IR), its harmonic extension to the upper halfplane is given by u(x, t) = Pt * f(x), where Pt is the Poisson kernel. We can also write (see (1.29))
'00
, u(z) where z
= x + it.
=
(
Jo
j(~)e27riZ'f, d~ + (
0
Joo
j(~)e27riz.f, d~,
If we now define
iv(z)
= (00 j(~)e27riz.f, d~ _ (0 j(~)e27riz.f, d~,
Jo
Joo
IRt
then v is also harmonic in and both u and v are real if f is. Furthermore, u + iv is analytic, so v is the harmonic conjugate of u. Clearly, v can also be written as
v(z)
=
k_isgn(~)e27rtlf,lj(~)e27rix.f, d~,
which is equivalent to
(3.1)
v(x, t) = Qt
* f(x),
where
(3.2) If we invert the Fourier transform we get
(3.3)
49
3. The Hilbert Transform
50
the conjugate Poisson kernel. One can immediately verify that Q(x, t) = Qt(x) is a harmonic function in the upper halfplane and the conjugate of the Poisson kernel Pt (x). More precisely,
Pt(x) which is analytic in 1m Z
. 1 t + ix + 2Qt(X) = 1ft  2 2 +x
i
= , 1fZ
> o.
In Chapter 2 we studied the limit as t  t 0 of u( x, t) using the fact that {Pt } is an approximation of the identity. We would like to do the same for v(x, t), but we immediately run into an obstacle: {Qt} is not an approximation of the identity and, in fact, Qt is not integrable for any t > o. Formally, 1 lim Qt(x) =  ;
t+O
1fX
this is not even locally integrable, so we cannot define its convolution with smooth functions.
2. The principal value of l/x We define a tempered distribution called the principal value of viated p. v. l/x, by
1
1 . ¢(x) p.v.(¢)=hm dx, x E+O jXj>E X
To see that
~his
p. v.
l/x,
abbre
¢ES.
expression defines a tempered distribution, we rewrite it as
r
~(¢) = x
Jjxj
¢(x)  ¢(O) dx + x
Jjxj>l
this holds since the integral of l/x on E < that p. v.
r
Ixl < 1 is zero.
¢(x) dx; x It is now immediate
~(¢) < C(II¢'lloo + Ilx¢lloo). x
,. 1 1 Proposition 3.1. In S , hm Qt =  p. v. . t+O
X
1f
Proof. For each E > 0, the functions 'l/JE(X) = X1X{jxj>E} are bounded and define tempered distributions. It follows at once from the definition that in
S' , 1 lim'l/JE = p. v. . E+O x Therefore, it will suffice to prove that in S'
lim (Qt 
t+O
~'l/Jt)
1f
=
O.
51
3. The theorems of M. Riesz and Kolmogorov
Fix ¢ E 5; then (nQt  'l/;t)(¢)
=
r ~¢(x)2 dx  r
+x = r ~¢(x)2 J1xl
¢(x) dx x
J1xl>t
dx
+
_r
r (2t + x 2 ~) ¢(x) dx x r ¢(tx) d X
J1xl>t
x¢(tx) d _  J 1xl<1 1 + x 2 x J 1xl>1 x(l
+ x 2)
X.
If we take the limit as t 7 0 and apply the dominated convergence theorem, we get two integrals of odd functions on symmetric domains. Hence, the limit equals O. D
As a consequence of this proposition we get that lim Qt t~O
* f(x) = .!.lim n
E~O
1
f(x  y) dy,
IYI>E
y
and by the continuity of the Fourier transform on 5' and by (3.2) we get
Given a function f E 5, we define its Hilbert transform by anyone of the following equivalent expressions:
Hf = t70 limQt * f, Hf
1
=
1
p. v. 
* f,
x (Hfr(~) = isgn(Oj(~)· n
The third expression also lets us define the Hilbert transform of functions in L2(~); it satisfies
(3.4) (3.5)
IIH fl12 = Ilflb,
(3.6)
!Hf'9=!f.H9.
H(Hf)=f,
3. The theorems ofM. Riesz and Kolmogorov The next theorem shows that the Hilbert transform, now defined for functions in 5 or L2, can be extended to functions in LP, 1 < P < 00. Theorem 3.2. For f E
S(~)7
the following assertions are true:
52
3. The Hilbert Transform
(1) (Kolmogorov) H is weak (1,1):
I{x
E
C
IR: IHf(x)1 > A}I < ~IIfI11.
(2) (M. Riesz) H is strong (p,p), 1 < p <
00:
IIH flip < Cpllfllp· Proof. (1) Fix A > 0 and f nonnegative. Form the Calder6nZygmund decomposition of f at height A (see Theorem 2.11); this yields a sequence of disjoint intervals {Ij } such that
f(x) < A for a.e. x tj.
n = UI j , j
Inl < 1.1 A < _1 IJ
1
A"f"1'
J, f < 2A. Ij
Given this decomposition of IR, we now decompose functions, g and b, defined by g
(x) = { f 1(x)J IIj I
if x tj.
f
as the sum of two
n,
. Ij f If x E I j ,
and
b(x)
=
L
bj(x),
j
where
bj(x) = (f(X)
~ I~I J,; f) Xlj(X)
Then g(x) < 2A almost everywhere, and bj is supported on I j and has zero integral. Since H f = H g + Hb,
I{x E IR: IHf(x)1 > A}I
< I{x
E IR :
IHg(x)1 > A/2}1
+ I{x E IR : IHb(x)1 > A/2}1·
We estimate the first term using (3.4):
I{x E ffi.:
IHg(x)1 > ,\/2}1 <
Gr
l1H9(xWdX
= : 2 l g(x)2 dx <
~l
g(x) dx
53
3. The theorems of M. Riesz and Kolmogorov
=
8
rj(x) dx.
A JlR
Let 2Ij be the interval with the same center as I j and twice the length, and let 0* = Uj 2Ij . Then 10*1 < 2101 and
I{x E IR:
IHb(x)1 >
A/2}1 < 10*1 + I{x rf. 0* : IHb(x)1 > A/2}1 <
~lljlll + ~ A
r
A JlR\o*
IHb(x)1 dx.
Now IHb(x)1 < Lj IHbj(x) I almost everywhere: this is immediate if the sum is finite, and otherwise it follows from the fact that L bj and L Hbj converge to band Hb in L2. Hence, to complete the proof of the weak (1,1) inequality it will suffice to show that
Even though bj
rf. S,
when x
rf. 2Ij
Hbj(x) =
the formula
r bj(y) dy
JI XJ
Y
is still valid. Denote the center of I j by ej; then since bj has zero integral,
The last inequality follows from the fact that Ix  ej 1/2. The inner integral equals 2, so
Iy  ejl < IIjl/2 and Ix  yl >
Our proof of the weak (1,1) inequality is for nonnegative j, but this is sufficient since an arbitrary real function can be decomposed into its positive and negative parts, and a complex function into its real and imaginary parts.
3. The Hilbert Transform
54
(2) Since H is weak (1,1) and strong (2,2), by the Marcinkiewicz interpolation theorem we have the strong (p,p) inequality for 1 < p < 2. If p > 2 we apply (3.6) and the result for p < 2:
IIHIii p
= sup {
= sup {
L
1} LI· Hg : IIgll", < 1} HI· 9
:
iigiip' <
< IIIllpsup{IIHgllp' : Ilgllpl < I} < Cp/ilillp,
o The functions 9 and b in the proof of the first part of this theorem are traditionally referred to as the good and bad parts of I. It follows from these proofs that the constants in the strong (p,p) and (p', p') inequalities coincide and tend to infinity as p tends to 1 or infinity. More precisely,
Cp = O(p)
as
p
and
* 00,
Cp= O((p  1)1)
as
p * 1.
By using the inequalities in Theorem 3.2 we can extend the Hilbert transform to functions in LP, 1 < p < 00. If I E Ll and {In} is a sequence of functions in S that converges to I in Ll (i.e. lim IIIn  Ilh = 0), then by the weak (1, 1) inequality the sequence {HIn} is a Cauchy sequence in measure: for any E > 0, lim I{x m,n>oo
E ffi. :
I(H In  HIm)( X ) I > E} I = 0,
.
i
Therefore, it converges in measure to a measurable function which we define to be the Hilbert transform of I, If I E LP, 1 < p < 00, and {In} is a sequence of functions in S that converges to I in LP, by the strong (p,p) inequality, {HIn} is a Cauchy sequence in LP, so it converges to a function in LP which we call the Hilbert transform of I.
In either case, a subsequence of {H In}, depending on I, converges pointwise almost everywhere to HI as defined. The strong (p,p) inequality is false if p = 1 or p = seen if we let I = X[O,lj' Then
HI(x)
=
1 x log   , 7r
xI
and HI is neither integrable nor bounded.
00;
this can easily be
4. Truncated integrals and pointwise convergence
55
In the Schwartz class it is straightforward to characterize the functions whose Hilbert transforms are integrable: for ¢ E S, H ¢ E L1 if and only if J ¢ = O. We leave the proof of this as an exercise for the reader. For more information on the behavior of the Hilbert transform on L1, see Section 6.5. In Chapter 6 we will examine the space of integrable functions whose Hilbert transforms are again integrable, and the space (larger than LIX)) in which we can define Hf when I is bounded. (Also see Section 6.6.)
4. Truncated integrals and pointwise convergence For E > 0, the functions y1 X{iyi>E} belong to Lq(JR}, 1 functions
HEI(x) = ~
r
<
q
<
00,
so the
f(x  y) dy
7r JiYi>E
Y
are well defined if I E LP, P > 1. Moreover, HE satisfies weak (1,1) and strong (p,p) estimates like those in Theorem 3.2 with constants that are uniformly bounded for all E. To see this, we first note that (~) =
! ·!
lim
N,;IX)
=
E
l1m
N,;IX)
=
e27riye Y
.
~
E
2i sgn(~) lim N ';IX)
dy
sin(27ry~) dy Y
27rN'~' sin(t) dt. 127rEI~1 t
This is uniformly bounded, so the strong (2,2) inequality holds with constant independent of E. We can now prove the weak (1,1) inequality exactly as in Theorem 3.2, and the strong (p, p) inequalities follow by interpolation and duality.
If we fix I E LP, 1 < p < 00, then the sequence {HEI} converges to HI as defined above in LP norm if p > 1 and in measure if p = 1. To see this, fix a sequence {In} converging to I in LP. Then
HI = n,;IX) lim H In = lim lim H(In = lim lim HEIn = lim HEI; n,;IX) E';O E';O n,;IX) E';O the second and third equalities hold because of the corresponding (uniform) (p, p) inequality. We now want to show that the same equality holds pointwise almost everywhere.
3. The Hilbert Transform
56
Theorem 3.3. Given
f
E
LP, 1
00,
Hf(x) = lim Htf(x)
(3.7)
t40
then a.e. xEIR.
Since we know that (3.7) holds for some subsequence {Htkf}, we only need to show that lim Htf(x) exists for almost every x. By Theorem 2.2 (and the remarks following it) it will suffice to show that the maximal operator
H* f(x) = sup IHtf(x)1 t>O
is weak (p, p). This however, follows from the next result. Theorem 3.4. H* is strong (p,p), 1 < p <
00,
and weak (1,1).
To prove this we need a lemma which is referred to as Cotlar's inequality.
< M(Hf)(x) + CMf(x).
Lemma 3.5. If f E S then H* f(x)
Proof. It will suffice to prove this inequality for each HE with a constant independent of E.
Fix a function ¢ E S(IR) which is nonnegative, even, decreasing on (0,00), supported on {x E IR : Ixl < 1/2} and has integral 1. Let ¢t(x) = E 1¢(X/E). Then
(cp, * p. v. ~) (Y) + [;X{lYI>'}  (cp, * p. v. ~) (Y)] ,
;X{lYI>'} =
and the convolution of the first term on the righthand side with f is dominated by M(H f) (x) (d. Proposition 2.7). It will s,uffice to find the pointwise estimate for the second term when E = 1 since it follows for any other E by dilation. If
Iyl > 1 then 1
r
y J 1
¢(x) d xl<1/2 Y  x x
=
1
Ixl<1/2
r
<
 Jxl<1/2 1
¢(x)
(1 1)   Y Y x
¢(x)lxl Iylly  xl
dx
dx
C
<2; y
if
Iyl < 1 then ·  11m 840
1 Ixl>8
¢(y  x) d x < x
r
Jxl<2 1
¢(y  x)  ¢(y) dx < C. x
57
4. Truncated integrals and pointwise convergence
Hence,
(I>, * p. v.
;X{IYI>'l 
D
(y) < 1
~ y2'
and by Proposition 2.7 the convolution of the righthand term with dominated by Mf(x).
f
is 0
Proof of Theorem 3.4. Since both the maximal function and the Hilbert transform are strong (p,p), 1 < p < 00, it follows at once from Lemma 3.5 that H* is strong (p, p). To show that H* is weak (1,1), we argue initially as in the proof of Theorem 3.2. We may assume that f > O. Now fix A > 0 and form the Calder6nZygmund decomposition of f at height A. Then we can write f as
f
= 9
+b= 9+ L
bj
.
j
The part of the argument involving 9 proceeds as in Theorem 3.2 using the fact that H* is strong (2,2). Therefore, the problem reduces to showing that
I{x rf. 0* Fix x holds:
rf.
0*,
E
: H*b(x) >
A}I < ~ Ilblh·
> 0 and bj with support I j . Then one of the following
(1) (xE,x+E)nIj=Ij ,
(2) (x  E, X + E) n I j = 0, (3) x  E E I j or x + E E I j . In the first case, HEbj(x) = O. In the second, HEbj(x) = Hbj(x); hence, if we let Cj denote the center of Ij , since bj has zero average,
IIjl Ibj(y)1 dy < I 1211bjlll. lj X  Y x  Cj x  Cj In the third case, since x rf. 0*, I j C (x  3E, X + 3E), and for all y Ix  yl > E/3. Therefore, 1
1
IHEbj(x) I < /, ~ 
r ibj{y)11 dy < ~ r xy
lH,bj{x) I :s
X
If we sum over all j's we get
IHEb(x)1 < L. Ix J
+3,
Ibj{y)1 dy.
E }X3E
}lj
31
11 .1 _J
.1211bjlll + E c J
x
+3E
X3E
IIjl  ~ Ix _ Cjl211bjlll + CMb(x).
<
J
Ib(y)1 dy
E Ij ,
3. The Hilbert Transform
58
. It follows from this that
I{x
+ I{x E JR.: Mb(x) > '\/2G}1 <
~llbjlh" r Ix IIjlCj 12 dx + ~'lIblh /\ 7 JlR\2Ij /\
G" < Tllbl1 1 .
o 5. Multipliers Given a function m E Loo(JR.n ), we define a bounded operator Tm on L2(JR.n ) by
(3.8)
(Tmff(~) = m(~)j(~).
By the Plancherel theorem, Tmf is well defined if f E L2 and
IITmfl12 < Ilmll oo llfl12. It is easy to see that the operator norm of Tm is IImll oo : fix E > 0 and let A be a measurable subset of {x E JR.n : Im(x)1 > Ilmll oo  E} whose measure is finite and positive, and let f be the L2 function such that j = XA. Then
We say that m is the multiplier of the operator Tm , though occasionally we will refer to the operator itself as a multiplier. When Tm can be extended to a bounded operator on LP we say that m is a multiplier on LP. For example, the multiplier of the Hilbert transform, m(~) = i sgn(~), is a multiplier on LP. More generally, given a, bE JR., a < b, define ma,b(~) = X(a,b) (0· Let Sa,b be the operator associated with this multiplier:
(Sa,bff (~)
=
X(a,b) (~)j(~).
We have the equivalent expression (3.9) where Ma is the operator given by pointwise multiplication by e27fiax,
Maf(x)
= e27fiax f(x).
To see this, note that by (1.16) the multiplier of iMaHMa is sgn(~  a). Ma is clearly bounded on LP, 1 < P < 00, with norm 1. Therefore, from (3.9) and the strong (p,p) inequality for H, we get that Sa,b is bounded on
5. Multipliers
59
LP. By making the obvious changes in the argument we see that this is still true if a = 00 or b = 00. Hence, we have the following result. Proposition 3.6. There exists a constant C p, 1 < p < a and b, 00 < a < b < 00,
00,
such that for all
For an application of this result, let a = R, b = R. Then Sa ,b is the partial sum operator SR introduced in Chapter 1: SRf = DR * f, where DR is the Dirichlet kernel. Hence,
IISRfll p < Cpllfllp with a constant independent of R. This yields the following corollary. Corollary 3.7. If f E LP(JR), 1
00,
then
lim IISRf  flip RHX)
When p
= 1 we do
= O.
not have convergence in norm but only in measure:
lim I{x E JR :
R'>oo
ISRf (x) 
f (x) I > E} I = O.
From this result and from Corollary 3.7 we see that there exists a sequence {SRkf(x)} that converges to f(x) for almost every x, but the sequence depends on f. Given a family of uniformly bounded operators on LP, any convex combination of them is also bounded. Hence, starting from Proposition 3.6 we can prove another corollary. Corollary 3.8. If m is a function of bounded variation on JR, then m is a multiplier on LP, 1 < p < 00. Proof. Since m is of bounded variation, the limit of m(t) as t 4 00 exists, so by adding a constant to m if necessary we may assume that this limit equals O. Furthermore, we may assume m is normalized so that it is right continuous at each x E JR. Let dm be the LebesgueStieltjes measure associated with m; then
m(~) = j€ 00
dm(t) = ( X(oo,.;)(t) dm(t) = (
J~
J~
X(t,oo)(~) dm(t).
Therefore,
(Tmff(~) = faX(t,oo)(~)j(~) dm(t),
3. The Hilbert Transform
60
and so
Tmf(x)
=
l St,=f(x) dm(t).
By Minkowski's inequality
IITmfilp < lIISt,=fll p Idml(t) < Cpllfllp lldml(t); the integral of finite.
Idml
is the total variation of
m and
by assumption this is D
Given a multiplier, we can construct others from it by translation, dilation and rotation. Proposition 3.9. Ifm is a multiplier on LP(JRn ), then the functions defined by m(~ + a), a E JR n , m()..~), ).. > 0, and m(p~), p E O(n) (orthogonal transformations), are multipliers of bounded operators on LP with the same norm as Tm. The proof of this result follows at once from properties (1.16), (1.17) and (1.18) of the Fourier transform. If m is a multiplier on LP(JR), then the function on JRn given by m(~) = m(6) is a multiplier on LP(JRn ). In fact, if Tm is the onedimensional operator associated with m, then for f defined on JR n ,
Tmf(x)
=
TmfC X2,· .. ,Xn)(Xl).
Then by Fubini's theorem and the boundedness of Tm on LP(JR),
r
jITln
ITmf(x)IP dx
J. (jITlr < J. J.
=
ITlnl
C
ITlnl
ITl
ITmfC X2,' .. ,xn)(xl)IP dX 1 ) dX2'" dX n If(Xl,'" ,xn) IP dXl ... dxn·
If we take m = X(O,=), then by Proposition 3.6 m is a multiplier on LP(JR) , 1 < p < 00. Hence, by the preceding argument the characteristic function of the halfspace {~ E JRn : 6 > O} will be a multiplier on LP(JRn ). Further, by Proposition 3.9 the same will be true for the characteristic function of any halfspace (since it can be gotten from the one above by a rotation and translation). The characteristic function of a convex polyhedron with N faces can be written as the product of N characteristic functions of halfspaces, so it is also a multiplier of LP(JRn ), 1 < p < 00. This fact has the following consequence.
Corollary 3.10. If P then
c JRn is a convex polyhedron that contains the origin,
lim IISAPf  flip = 0,
.A.>=
1
00,
6. Notes and further results
61
where S>.p is the operator whose multiplier is the characteristic function of
AP = {AX: X E P}. In light of the above observation, it is perhaps surprising that when n > 1, the characteristic function of a ball centered at the origin is not a bounded multiplier on LP(lRn ) , p I 2. We will examine this and related results for multipliers on lR n in Chapter 8. (Also see Section 6.8 below.)
6. Notes and further results 6.1. References.
The proof of the theorem of M. Riesz first appeared in Sur les fonctions conjuguees (Math. Zeit. 27 (1927), 218244) but had been announced earlier in Les fonctions conjuguees et les series de Fourier (C. R. Acad. Sci. Paris 178 (1924), 14641467). The proof in the text became possible only after the Marcinkiewicz interpolation theorem appeared in 1939. See Section 6.3 below for the original proof by Riesz and another one due to M. Cotlar (A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105167). Additional proofs due to A. P. Calderon (On the theorems of M. Riesz and Zygmund, Proc. Amer. Math. Soc. 1 (1950), 533535) and P. Stein (On a theorem of M. Riesz, J. London Math. Soc. 8 (1933), 242247) are reproduced in Zygmund [22]. Kolmogorov's theorem is in Sur les fonctions harmoniques conjuguees et les series de Fourier (Fund. Math. 7 (1925), 23~28). Another proof due to L. H. Loomis (A note on Hilbert's transform, Bull. Amer. Math. Soc. 52 (1946), 10821086) can be found in Zygmund [22] and an elegant proof due to L. Carleson is given by Katznelson [10]. Our proof uses the CalderonZygmund decomposition in the same way that it appears in Chapter 5 for more general singular integrals. It is derived from the proof of A. P. Calderon and A. Zygmund (On the existence of certain singular integrals, Acta Math. 88 (1952), 85139). Theorem 3.5 is in the paper by M. Cotlar cited above. 6.2. The conjugate function and Fourier series.
The results in this chapter can also be developed for functions defined on the unit circle. Historically, they evolved in close connection with the theory of Fourier series discussed in Chapter 1. Given a function f E L1(1I') (equivalently, a function in L1([0, 1])), its harmonic extension to the unit disk is gotten by convolution with the Poisson kernel
P (t) r

1  r2 1  2r cos(2nt)
+ r2 '
0 < r < 1,
3. The Hilbert Transform
62
and the harmonic conjugate of Pr conjugate Poisson kernel
t _ Qr ()
* f(t)
is gotten by convolution with the
2r sin(27rt) 1  2r cos(27rt) + r2 '
O
The analogue of the Hilbert transform is the conjugation operator which is defined by (3.10)

f(x) = lim Qr r71
* f(x) = E70 lim
1
f(x  t)  f(x () tan 7rt
E
+ t)
dt.
Both limits can be shown to exist for almost every x. The theorems of M. Riesz and Kolmogorov hold for' the conjugation operator. Hence, by an argument similar to the one in Section 5 it can be shown that the partial sum operators S N f of the Fourier series of fare uniformly bounded on LP, and so S N f converges to f in V. (Cf. Chapter 1, Section 4.) Given a function fELl (1['), its conjugate Fourier series is the trigonometric series 00
S[J](x)
=
L i sgn(k)j(k)e27rikx; 00
when the Fourier series of jis written as in (1.1), then the conjugate series is gotten by switching the coefficients of the sine and cosine terms and taking their difference instead of their sum: 00
S[J](x)
=
L
ak sin(27rkx)  bk cos(27rkx).
k=O
When JELl then S[J] coincides with the Four~er series of
j.
Even for continuous functions the second limit in (3.10) exists because of subtle cancellation and not because the numerator gets small for t close to zero. It can be shown using the Baire category theory that there exists a continuous function f such that for every x E [0,1]'
r
1/ 2
10
If(x  t)  f(x
+ t)1
dt
= 00.
tan(7rt) The same phenomenon occurs with the Hilbert transform. For these and further results, see the books by Bary [1], Katznelson [10], Koosis [11] and Zygmund [21]. 6.3. Other proofs of the M. Riesz theorem.
The original proof of M. Riesz of the LP boundedness of the conjugate function is based on the analyticity of powers of an analytic function. The easiest case is when p = 2k, k an integer. Given a function f on the unit
6. Notes and further results
63
circle, let u = Pr * f and v = Qr * f; then F(z) = u + iv is analytic in the unit disc and hence so is F(z)2k. By Cauchy's theorem
~ 27r'l
r F(z)2k dz J1z1=r z
=
21f F(re r 27r J o
~
it )2k
dt
=
F(0)2k.
If F(O) = 0 then by taking the real part 9f F(re it ) we get that
21f r Iv(re Jo
it )
12k dt <
t
j=l
(~~) ro21f lu(re J J
it ) 12j
Iv(re it ) 12k  2j dt.
By applying Holder's inequality (with the appropriate exponent) to each term on the righthand side, we get
fo2~ Iv(re;t) 12k dt < Ckfo2~ lu(reit ) 12k dt, where C k is independent of the radius of the circle. Taking the limit as the radius tends to 1, u tends to f and v to j. (When F(O) = j(O) =I 0 replace f by f  j(O).) From the inequality for p = 2k, the whole theorem can be obtained using interpolation and duality, but since interpolation was not available in 1923, Riesz next considered the case when p > 1 is not an integer. In order to get that (u+iv)P is analytic, he assumed u > 0 (which is true if f is nonnegative and not identically zero) and then used a clever estimate. For technical reasons, this argument does not work for p an odd integer, and the inequality for these values of p is gotten by duality. In a letter dated November 1923, Riesz explained to Hardy how he arrived at the proof and showed him the essential details (reproduced in M. L. Cartwright, Manuscripts of Hardy, Littlewood, M. Riesz and Titchmarsh, Bull. London Math. Soc. 14 (1982), 472532). From the fact that (u + iv)2 = u 2  v 2 + 2iuv is analytic we deduce H(f2  (H J)2) = 2f . H f, which gives the following formula due to Cotlar:
(3.11)
(H f)2
= f2 + 2H(f . H f).
An alternative way to prove this (proposed by Cotlar in his paper) is to show directly that
HI * HI = j * j 
2isgn(~)(j * HI). It follows from (3.11) that if IIH flip < Cpllfll p,' then IIH fl12p < (Cp +
JC~ + 1)llfI12p.
In fact,
IIHfll~p
=
II(Hf)21Ip
< IIf211p + 211 H (f· Hf)llp
< Ilfll~p + 2Cpllf . H flip
3. The Hilbert Transform
64
and the desired inequality follows by solving a quadratic equation. If we begin with (3.4) and repeatedly apply this inequality we get, for instance, that IIH fl12k < (2k  1) Ilflbk for integers k > 1. By the RieszThorin interpolation theorem (Theorem 1.19), we get
We get the Riesz theorem for p Theorem 3.2.
< 2 by using duality as in the proof of
An elementary realvariable proof when p transform) is the following:
= 2 (without using the Fourier
and the kernel of HEHE, which is the convolution of y1 X {IYI>E} with itself, can be computed explicitly and shown to be in L1. (By a dilation argument, it suffices to consider the case E = 1.) This proof is attributed to N. Lusin (1951) by E. M. Dyn'kin (Methods of the theory of singular integrals: Hilbert transform and Calder6nZygmund theory, Commutative Harmonic Analysis, I, V. Havin and N. Nikolskii, eds., pp. 167259, SpringerVerlag, Berlin, 1991). A similar argument had been used earlier by Schur for a discrete version of the operator. Another realvariable proof when p ter 9, Section 1.
=
2 using Cotlar's lemma is in Chap
6.4. The size of constants. The best constants in the theorems of Kolmogorov and M. Riesz are known. For the strong (p,p) inequality, 1 < p < 00, the best constants are Cp = tan(7r/2p), 1 < p < 2, and Cp = cot (7r/2p) , 2 < p < 00. This was first proved by S. K. Pichorides (On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165179); a much simpler proof was later given by L. Grafakos (Best bounds for the Hilbert transform on LP(Il.~l), Math. Res. Let. 4 (1997),469471). In the proof of the M. Riesz theorem due to Cotlar given above, we get
J
the estimate C2p < Cp + C; + 1. If we begin with C2 = 1, the resulting bounds for C 2 k are sharp. This fact was first observed by 1. Gohberg and N. Krupnik (Norm of the Hilbert transformation in the space LP, Funct. Anal. Appl. 2 (1968), lSDlSI).
6. Notes and further results
65
The best constant in the weak (1,1) inequality has a more complex expression:
This was first proved by B. Davis (On the weak type (1,1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307311) using Brownian motion. A nonprobabilistic proof was later given by A. Baernstein (Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833852).
6.5. The Hilbert transform on LI. As we noted in Section 3, if fELl then H f need not be. However, unlike the case of the maximal function, there exist fELl such that H fELl. For example, if f = X(O,I)  X(I,O) then Hf(x) = log(lx 2  11/x 2 ) is integrable. A simple necessary condition for H f to be integrable if fELl is that j(O) = J f = o. If f is such that the identity (H ff(~) = i sgn(Oj(~) holds, then it suffices to note that the Fourier transform of an integrable function is continuous, and j(~) and i sgn(~)j(~) are both continuous only if j(O) = O. This argument clearly works if fELl n L2 since in this case the Fourier transform identity is valid. But one can show that this identity holds if f, Hf ELI. This would follow if ¢ * (Hf) = H(¢ * f) for every ¢ E S: take the Fourier transform of both sides and simplify. This equality can be proved as is done for more general singular integrals in the paper by A. P. Calderon and O. Capri (On the convergence in LI of singular integrals, Studia Math. 78 (1984),321327). Similarly, when f, H fELl we have the identity H(H f) = f. We originally gave this in (3.5) for L2 functions, and it can be extended to LP, 1 < p < (Xl, using the results in this chapter. For f and H f integrable, a proof using complex analysis is due to E. Hille and J. D. Tamarkin (On the absolute integrability of the Fourier transform, Fund. Math. 25 (1935), 329352). The interesting paper by J. F. Toland (A few remarks about the Hilbert transform, J. Funct. Anal. 145 (1997), 151174) examines this question and provides additional historical information. In general, H f ¢:. LI, so we cannot even define the Hilbert transform of H f. Nevertheless, if we replace the Lebesgue integral with a more general one, then it is possible to define H(H f) and to get the identity. (See Zygmund [21, Chapter 7].) The abovecited paper by Toland uses this extended notion of the integral.
3. The Hilbert 'ITansform
66
6.6. L log L estimates.
The following result characterizes the local integrability of the Hilbert transform; it is analogous to Theorem 2.18 for the maximal function. Theorem 3.11. Let Band C be two open balls such that
(1) If flog+ If I E L1(C), then Hf E L 1(B). (2) Conversely, if f > 0 and Hf E L1(C), then flog+
Bc
C.
If I E L1(B).
The first part of the theorem is due to A. P. Calderon and A. Zygmund (see the paper cited above). It can be sharpened using Orlicz spaces (cf. Chapter 2, Section 8.4, and Zygmund [21, Chapter 4]). The second part is due to E. M. Stein (Note on the class L log L, Studia Math. 32 (1969), 305310). The hypothesis f > 0 can be replaced by a weaker condition which measures (in some sense) the degree to which Pt * f differs from a positive harmonic function. See the article by J. Brossard and L. Chevalier (Classe LlogL et densite de l'integrale d'aire dansIR~+\ Ann. of Math. 128 (1988), 603618).
6.7. Translation invariant operators and multipliers. Let Th be the translation operator: Thf(x) = f(x  h). An operator T is said to be translation invariant if it commutes with Th: Th 0 T = To Th, h E IRn. The Hilbert transform, for example, is translation invariant. Such operators are completely characterized by the following result. Theorem 3.12. Given a linear operator T which is translation invariant and bounded from LP(IRn) to Lq(IRn ) for any pair (p, q), 1
If such a T is bounded from LP to Lq and not zero, then p cannot be greater than q. An elegant and very simple proof of this fact is due to L. Hormander (Estimates for translation invariant operators in LPspaces, Acta Math. 104 (1960), 93139). First note that limh~= Ilf + Thfll p = 21/ Pllfll p· Let IITllp,q be the norm of T as a bounded operator from LP to Lq. Since T commutes with translations,
liT f + ThT fllq <
IITllp,q Ilf
+ Thfll p;
if we let h tend to infinity we get
IITfllq < 21/P1/qIITllp,qllfllp, and this is possible only when p < q. Clearly, if T f = K * f then T is linear and translation invariant. By the properties of the Fourier transform, (T ff = i( j, which implies that k is a
67
6. Notes and further results
multiplier of T, at least if K E L=, following the definition in Section 5. But if T is bounded on LP, then by duality (since its adjoint is the convolution with kernel K(x) = I«x)) it is bounded on LP'. Thus by interpolation it is bounded on L2, so the condition that k E L= is necessary, and our assumption that a multiplier m is in L= is justified. It would be interesting to characterize the distributions K that define bounded operators (equivalently, to characterize the multipliers on LP), but a complete answer is known only in two cases.
Proposition 3.13.
(1) T is bounded on L2(IRn) if and only if k E L= and the norm of T as an operator on L2 is the norm of k in L=. (2) T is bounded on L1(IRn) if and only if K is a finite Borel measure, and the operator norm ofT equals the total variation of the measure.
Note that since p. v. l/x is not a measure, Proposition 3.13 gives another proof that the Hilbert transform is not bounded on L1. For all of these results, see Stein and Weiss [18, Chapter 1]. 6.8. Multipliers of Fourier series.
The multipliers of trigonometric series are precisely bounded sequences, {.\( k) hEZ. The associated operator is defined for f E L2 (1I') by
T)J(x) =
L
'\(k)j(k)e27fikx.
kEZ
(For multiple Fourier series the definition is the same except that we take k E 7l n .) T>.. can be written as the convolution of f with a distribution whose Fourier coefficients are {.\ (k )}, and there are results analogous to those in Proposition 3.13.
If we know the boundedness properties of a multiplier on LP(IR), then we can deduce results for multipliers of series by restricting the multiplier to the integers. Theorem 3.14. Given 1 < P < 00, suppose the operator T defined by T f = K * f is bounded on LP(IR). If k is continuous at each point of 7l and .\(k) = k(k), then the operator T>.. associated with this sequence is bounded on LP(1I') and liT>.. I < IITII.
There is also an analogous ndimensional result. For a proof see Stein and Weiss [18, Chapter 6]. Starting from Theorem 3.14, one can deduce results for the partial sum operators SN of a Fourier series from the corresponding results for SR in
68
3. The Hilbert Transform
Section 5, and in particular one gets the convergence in LP norm of the partial sums of a Fourier series. (Cf. Chapter I, Section 4.) 6.9. The Hilbert transform of characteristic functions. If A is a subset of IR with finite measure, one can show that
21AI
I{x E IR: IH(XA)(x)1 > A}I = sinh(rrA)" This result was first proved by E. M. Stein and G. Weiss (An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959),263284). A simple proof is given by Zygmund [22, p. 15].
4
Chapter
Singular Integrals (I)
1. Definition and examples
The singular integrals we are interested in are operators of the form
Tf(x) = lim
(4.1)
10>0
r 0 y2f(x  y) dy, J1Y1>f (
1
Y
1
where 0 is defined on the unit sphere in IR n , snl, is integrable with zero average and where y' = Y/IYI. With these hypotheses, (4.1) is defined for Schwartz functions since it is the convolution of f with the tempered distribution p. v. O(x')/Ixl n , defined by
(4.2) p. v.
O(x')
1.
.
X InI (¢) = 10>0 hm =
O(x')
IX In
Ixl>f
¢(x) dx
r 0 x2 i1>(x) 1>(0)] dx + r ~(~21>(x) dx. J1xl<1 J1xl>1 1
(
X
1
X
(The second equality follows since 0 has zero average.) Since ¢ E S, both integrals converge. Note that we could also assume, for instance, that ¢ in (4.2) or f in (4.1) is a Lipschitz function of order a with compact support. Proposition 4.1. A necessary condition for the limit in (4.1) or (4.2) to
exist is that 0 have zero average on
snl.
Proof. Let f E S(IRn) be such that f(x)
Ixl < 1, Tf(x)
=
=
1 for any
Ixl <
2. Then for
r 0 y2f(x  y) dy + lim 1 0 y2dy. J1Y1>1 f
1
Y
(
1
1
10>0
y
1

69
4. Singular Integrals (I)
70
The first integral always converges but the second equals
0,(Y') dcr(y') . log(l/ E),
lim ( 1':;0 }
snl
and this is finite only if the integral of 0, on
snl
is zero.
D
When n = 1 the unit sphere reduces to two points, 1 and 1, and 0, must take opposite values on them. Thus on the real line, any operator of the type in (4.1) is a multiple of the Hilbert transform. In higher dimensions we consider two examples given by A. P. Calderon and A. Zygmund. First, let f(xl, X2) be the density of a mass distribution in the plane. Then its Newtonian potential in the halfspace IR.! is
U(XI' X2, X3)
=
J.
JR2
(Xl 
f(yl, Y2) ( )2 YI + X2  Y2
)2
+ x32 dyl dY2.
The strength of the gravitational field is gotten by taking partial derivatives of the potential. The component in the X3 direction is equivalent to a multiple of the Poisson integral which we have already considered. The other two are similar to one another; for example, for the first we formally have that
. au hm 8 (Xl, X2, X3)
X3;0
Xl
= 
1 JR2
[(
Xl
f(yl, Y2)(XI  YI) )2 + (X2  Y2 )2]3/2 dYldY2.  YI
This integral does not converge in general, but it exists as a principal value if f is smooth and is in fact the value of the limit of 8u/8xI. It corresponds to a singular integral of the form (4.1) in IR. 2 with 0,(x') = xI/lxl. (In polar coordinates this becomes cos( 8).) The second example is gotten from the logarithmic potential associated with a mass distribution f(xl, X2) in the plane (i'.e. the solution of the equation /:1u = f):
U(XI' X2)
=
(
JJR2
f(YI, Y2) log ([(
Xl  YI
)2
+1( X2
 Y2
)2]1/2) dYldY2·
This integral converges absolutely if, for example, f has compact support, and partial derivatives can be taken under the integral sign. However, this is not possible for second derivatives: one can show that 82u/8x18x2 is given by an operator of the form (4.1) with 0,(x') = 2xlx2/lxI2. For more information on this operator, see Section 5.
2. The Fourier transform of the kernel A function
f is homogeneous of degree
a if for any
X E
IR.n and any A > 0,
2. The Fourier transform of the kernel
Given any function cp, define cp)..(x)
=
71
).ncp().l x ); then
r f(x)cp)..(x) dx =).a JJRnr f(x)cp(x) dx,
J]Rn
so we can define the homogeneity of a distribution as follows.
Definition 4.2. A distribution T is homogeneous of degree a if for every test function cp it satisfies
A simple computation shows that the distribution given by (4.2) is homogeneous of degree n; details are left to the reader.
Proposition 4.3. If T is a tempered distribution which is homogeneous of degree a, then its Fourier transform is homogeneous of degree n  a. Proof. By Definition 1.16 and property (1.18), if cp E S then
T(cp)..) = T(¢()'.)) = ).nT(¢).._l) = ).naT(¢) = ).naT(cp).
o As a corollary to this proposition we can easily calculate the Fourier transform of f(x) = Ixl a if n/2 < a < n. For in this case f is the sum of an L1 function (its restriction to {Ixl < I}) and an L2 function, so by Proposition 4.3, f is a homogeneous function of degree a  n. Hence, since f is rotationally invariant, j(e) = ca,nlel a n . We calculate ca,n using Lemma 1.14 and (1.21): A
r
JJRn
A
e7rlxI2Ixla dx = ca,n
since
1
00
a
d=
7rr2 e rbr
J.JRn
e7rlxI2Ixlan dx;
1 r (1 + b)
7r
2
l+b 2

2'
we see that (4.3) In fact, this formula holds for all a, 0 < a < n. For 0 < a < n/2 it follows from the inversion formula (1.22). Further, since Ixl a tends to Ixl n/ 2 as a ) n/2, and since the Fourier transform is continuous, (4.3) holds with a = n/2.
72
4. Singular Integrals (I)
Theorem 4.4. If 0 is an integrable function on sn1 with zero average, then the Fourier transform of p. v. O(x')//x/ n is a homogeneous function of
degree
°
given by
m(E)
(4.4)
hn,
=
Q(u) [log
Cu lE'l)  i; sgn(u· E')1daru).
Proof. By Proposition 4.3 the Fourier transform is homogeneous of degree 0; therefore, we may now assume that /~/ = 1. Since 0 has zero average,
m(~) =
1
lim O(y') e27riy.~ dy E+O E
r
= lim
E+O}snl
Thus m(~)
+ 1,1/E e27riru.~dr] r
1
dcr(u).
h  ih, where
=
r
h = lim
O(u) [11 (e27riru.~ _ 1) dr E r
E+O}snl
O(u) [11 (cos (27rru E
.~) 
1) dr r
+ h .
=
1
lim O(u) E+O snl
/,'/< cos(21rru . 0 ~ ] datu),
[1
1/E sin(27rru· E
dr]
~)
r
dcr(u).
By the dominated convergence theorem we can exchange the limit and outer integral. In each inner integral make th~ change of variables s 27rr/u· ~/. We may assume that u· ~ # 0, so in 12 we get
/,
27rlu.~IE
As
E t
ds
27rlu.~I/E
0 this becomes sgn(u·~)
1= o
sin(s) sgn(u·
~). S
sin(s) ds s
7r 2
=  sgn(u· ~).
After the change of variables in h we get
ds 1,27r IU{I/E ds 1,27rIU.~1 ds cos(s) ; (cos(s)  1) + 27rlu·~IE s I S 1 s
/,
as
E t
1
0 this becomes
1 1
o
ds (cos(s)  1) +
j=
cos(s)
SIS
ds log /27r/log /u· ~/.
3. The method of rotations
73
If we integrate against 0, which has zero average on snl, the cop.stant terms disappear and we get the desired formula for m. 0 In formula (4.4) the factor multiplied against 0 has two terms: the first is even and its contribution is zero if 0 is odd; it is not bounded but any power of it is integrable. The second is odd and its contribution is zero if o is even; further, it is bounded. Since any function 0 on snl can be decomposed into its even and odd parts, 1
Oo(u) = "2(O(u)  O(u)),
we immediately get the following corollary.
Corollary 4.5. Given a function 0 with zero average on snl, suppose that 0 0 E Ll(snl) and Oe E Lq(snl) for some q > 1. Then the Fourier transform ofp. v. O(x')/Ixl n is bounded. From (4.4) one can easily find an integrable function 0 such that m is not bounded. Nevertheless, in Corollary 4.5 we can substitute the weaker hypothesis Oe E LlogL(snl), that is,
hnl
10e(u)llog+ 10e(u)1 dCJ(u)
< 00.
(Recall that log+ t = max(O, log t).) The sufficiency of this condition follows from the inequality
AB < A log A For in the region D
In
= {u
0.(u) log
<
In
E
+ eB ,
A > 1, B > O.
snl : 10(u)1 > I},
Cu CI) du(u) 1
210(u)llog(210(u)l) dCJ(u)
o is bounded in the complement of D,
+
In
lu· (1 1 / 2 dCJ(u).
and so the integral is finite.
3. The method of rotations Corollary 4.5 together with the Plancherel theorem gives sufficient conditions for operators T as in (4.1) to be bounded on L2. In this and the following section we develop techniques due to Calderon and Zygmund which will let us prove they are bounded on LP, 1 < p < 00. Let T be a onedimensional operator which is bounded on
LP(~)
and let
u E snl. Starting from T we can define a bounded operator Tu on ~n as follows: let Lu = {AU: A E ~} and let Lt be its orthogonal complement in ll~n. Then given any x E ~n, there exists a unique Xl E ~ and x E Lt such
4. Singular Integrals (I)
74
that x = XIU + X. Now define Tuf(x) to be the value at Xl of the image under T of the onedimensional function f( U + x). If Cpis the norm of T in LP(~), then by Fubini's theorem,
!.
ITuf(x)IP dx
=
F
<
(
JLt
!.
IT(f( U + x)(xl)IP dxldx
R
C: JLtr !. If( U + x)(xl)I Pdxldx C: JRnr If(x)IP dx. R
=
Operators gotten in this way include the directional HardyLittlewood maximal function,
Muf(x) = sup Ih h>O 2 and the directional Hilbert transform, . Huf(x) = 1 hm 7r
jh
If(x  tu)1 dt,
h
1,
E+O Itl>E
dt f(x  tu).
t
Since the operators Tu obtained from the operator T are uniformly bounded on LP (~n), any convex combination of them is also a bounded operator. Hence, the next result is an immediate consequence of Minkowski's integral inequality. Proposition 4.6. Given a onedimensional operator T which is bounded
on LP(IR) with norm Cp, let Tu be the directional operators defined from T. Then for any nELl (snl ), the operator Tn defined by Tnf(x) =
knl
n(u)Tuf(x) d(}(u)
is bounded on £p(IRn) .with norm at most cpllnlh. By using this result we can pass from a onedimensional result to one in higher dimensions. Its most common application is in the method of rotations, which uses integration in polar coordinates to get directional operators in its radial part. For example, given nELl (snl), define the "rough" maximal function
(4.5)
Mnf(x) = sup
R>O
r
IB(~, R)I JB(O,R) 1r2(y')lIf(x 
y)1 dy.
If we rewrite this integral in polar coordinates we get
Mnf(x) =
~~ IB(O,ll)IRn knl 1!1(u)1 fuR If(x 
ru) Irn  l drdo(u)
3. The method of rotations
r
< IB( 1 )1
75
Jsnl
0,1
10(u)IMuf(x) dO"(u).
Corollary 4.7. If 0 E L1 (sn1) then MD, is bounded on LP(IR(n)) 1 < p
<
00.
If we let O(u) = 1 for all u, then MD, becomes the HardyLittlewood maximal function in IR(n; thus the method of rotations shows it is bounded on V(IR(n), p > 1, starting from the onedimensional case. Further, it is interesting to note that the V constant we get is O(n) since Isn 1 1/IB(0, 1)1 = n. (See Chapter 2, Section 8.5.) On the other hand, the method of rotations does not give us the weak (1,1) result. (See Section 7.6.)
o
We can also apply Proposition 4.6 to operators of the form (4.1) when is odd. In fact, if we fix a Schwartz function f, then
T f (x)
!,snl 1 !,Snl 1.
dr f (x  ru)  dO" (u ) E r 00.
=
lim E>O
=
lim 1 E>O 2
0 (u )
O(u)
Irl>E
dr f(x  ru) dO"(u); r
since 0 has zero average, we can argue as we did in (4.2) to get =
~ lim
+~ 2
1 1.
r
O(u)
r
O(u)
2 E>O J Snl
JSnl
E
(f(x  ru)  f(x)) dr dO"(u) r
f(x  ru) dr dO"(u).
Irl>l
r
Because f E S, the inner integral is uniformly bounded, so we can apply the dominated convergence theorem to get =
7T
2
r
Jsnl
O(u)Huf(x) dO"(u).
Since the Hilbert transform is strong (p, p), 1 the following.
< p < 00, we have proved
Corollary 4.8. If 0 is an odd integrable function on snl) then the operator T defined by (4.1) is bounded on LP (IR(n)) 1 < p < 00. In Corollary 4.8 we implicitly defined T f for f E LP as a limit in LP norm. However, as with the Hilbert transform, we can show that (4.1) holds for almost every x. By an argument similar to the one above we can show that the maximal operator associated with the singular integral,
(4.6)
T* f(x) = sup E>O
r 0 y2f(x JIYI>E Y (
1
1
y) dy ,
4. Singular Integrals (I)
76
satisfies
T* f(x) <
7r
2
r
In(u)IH~f(x) d(/(u) ,
J
snl
where H~ is the directional operator defined from the maximal Hilbert transform. (See Chapter 3, Section 4.) Therefore, we can apply Proposition 4.6 and Theorem 2.2 to get the following.
Corollary 4.9. With the same hypotheses as before, the operator T* defined by (4.6) is strong (p, p), 1 < p < 00. In particular, given f E LP, the limit (4.1) holds for almost every x E jRn. An important family of operators with odd kernels consists of the Riesz transforms,
(4.7)
r
y.
Rjf(x) = Cn p. v. J~n IYI~+l f(x  y) dy,
where
1)
n+ cn=f ( 2The constant is fixed so that for
f
7r
_n+l 2
1 <j< n,
.
E L2,
(4.8) this in turn implies that n
""' 6 R~J =1 ' j=l
(4.9)
where I is the identity operator in L2. Since the Schwartz functions are dense in LP, p > 1, this identity in fact holds in every LP space. To prove (4.8) one could use Theorem 4.4 directly; instead we will argue as follows. With equality in the sense of distributions,
a I In+ 1 _ ax j x. so if we apply (4.3) we get
(
)
xj
1  n p. v. Ix In+ 1 '
77
4. Singular integrals with even kernel n+l .
=
7[2
.
'l, ;::;:
r (nt1)
~j
j~j.
4. Singular integrals with even kernel If the function D in the singular integral (4.1) is even, then the method of rotations does not apply since we cannot represent the singular integral in terms of the Hilbert transform. However, we ought to be able to argue as follows: by (4.9), n
Tf
n
LR](Tf) j=l
= 
LRj(RjT)f, j=l
= 
and the operator RjT is odd since it is the composition of an odd and even operator. If we can show that RjT has a representation of the form (4.1), then by Corollary 4.8, T is bounded on LP. In the rest of this section we will make this argument precise by showing that RjT has the requisite representation. Let D be an even function with zero average in Lq(sn1), for some q > 1, and for E > 0 let (4.10) Note that KE E L T , 1 < r < q. Thus if Fourier transform we see that
f
E
Cgo(JR n ) then by taking the
(4.11) Lemma 4.10. With the preceding hypotheses, there exists a function K j which is odd, homogeneous of degree n and such that
lim RjKE(x)
E~O
= Kj(x)
in the L oo norm on every compact set that does not contain the origin. Proof. Fix x i= 0 and let 0 by Corollary 4.9
RjKE(x)  RjKv(x)
< E < l/ < jxj/2. Then for almost every such x,
= Cn c5~O lim =C
J
J.
]Rn
j
Xj
X 
jY~lX{IXYI>c5}[KE(Y) Yn
Kv(Y)] dy
Xj  Yj D(y') dy· ' n E
since D has zero average, = Cn
J
E
(Xj  Yj Xj ) D(y') dy jx  yjn+1 jxjn+1 jyjn .
4. Singular Integrals (I)
78
Therefore, if we apply the mean value theorem to the integrand we get
(4.12)
C
IRjKE(x)  RjKv(x) 1< 1 In+l
1
x
10(y') 1 CIIOlh 1 Inl dy < 1 In+l v. x
E
Hence, for any a > 0, {RjKE} is a Cauchy sequence in the L= norm on { 1xl> a}. So for almost every x we can define by
K;
KJ~(x)
= E+O lim RjKE(x).
The function Rj KE is odd, so (by modifying necessary) is also an odd function.
K;
K; on a set of measure zero if
To find the desired function K j , fix A > 0; then again for almost every
x,
K;
K;
Hence, for almost every x, (AX) = An (x). The set of measure zero where equality does not hold depends on A, but since is measurable, the set
'D
=
{(x, A) C JRn
X
(0,00) : K;(AX)
K;
# An 1<; (x)}
has measure zero. Therefore, by Fubini's theorem there exists a sphere centered at the origin of radius p, Sp, such that D n Sp has measure zero. We now define '
Kj(x) =
UI:I) K; (I~)
if x
n
# 0 and px/lxl rt. D n Sp;
otherwise. This function is measurable, homogeneous of degree n (by definition) and odd. Further, K;(x) = Kj(x) almost everywhere. To see this, let x # 0 be such that Xo = px/lxl rt. D n Sp. (The set of x such that Xo E D n Sp has measure zero since D n Sp has measure zero.) Then for almost every A,
Kj(Axo) = AnKj(xo)
=
AnK;(xo) = K;(AXo).
o Lemma 4.11. The kernel K j defined in Lemma 4.10 satisfies
hnl
IKj(u)1 d(J(u) <
CqlIOll q·
4. Singular integrals with even kernel
Furthermore, if Kj,E(X) and II~EI11 < C~IIOllq·
= Kj(x)X{lxl>E}' then ~E = RjKE Kj,E
Proof. By the homogeneity of K j
r
IKj(u)1 dO"(u)
J
79
=
Snl
E
L1(JR n )
,
II/, og
2 1
1
IKj(x)1 dx
~
/,
< log2 1
+  12 1 /, og
=
In (4.12) let v
1/2 and Ixi
1
IRjK1/2(X)1 dx.
> 1; then if we take the limit as
~
E
~
0 we get
CII0111
IKj(x)  R j K 1/2(X)1 < Ixln+1·
(4.13)
Therefore, the first integral on the righthand side above is bounded by CII0111 < CIIOllq. To bound the second integral, note that
IRjK1/2(X)1 dx < CIIRj K 1/21Iq < CIIK1/ 2 1Iq < CIIOllq·
/,
1
To prove the second assertion it will suffice to show that II~llh since ~E = E n ~1 (E 1X). But then
II~Iil1 = <
r
IRj K 1(x)  K j,1(x)1 dx
r
IRj K 1(x)ldX+/,
JlR
<
00
n
Jxl<2
IKj(x)ldx+
1
1
r
J1xl>2
l~l(X)ldx.
The first integral is bounded by CIIRj K 111q < CIIK111q < CIIOllq; above we showed that the second integral has the same bound. To see that the third integral is bounded, we argue as we did for (4.13) to get l~l(X)1 < CII0Ihlxln1. 0 Theorem 4.12. Let 0 be a function on sn1 with zero average such that its odd part is in L1 (sn1) and its even part is in Lq (sn1) for some q > 1. Then the singular integral T in (4.1) is bounded on LP (JR n ) , 1 < p < 00. Proof. By Corollary 4.8 we may assume 0 is even. Further, by arguing as we did for the Hilbert transform, it will suffice to establish the LP inequality for functions f E Cgo(JR n ). For such f, Tf = limE_o KE * f. From (4.9) and (4.11) we see that n
I{E
*f = 
~Rj ((RjKE) * f), j=l
4. Singular Integrals (I)
80
and in the notation of Lemma 4.11,
By Lemma 4.10, K j is an odd, homogeneous kernel of degree n, so by Corollary 4.9,
II Kj,E * flip < C
!snl 'Kj(u)1 da(u)lIflip < CIIOllqllfll p·
By Lemma 4.11,
If we combine these estimates and use the fact that Rj is bounded in LP we see that
Since the righthand side is independent of
E,
by Fatou's lemma we get
o
and this completes our proof.
Another proof of Theorem 4.12 is given below in Chapter 8. (See Corollary 8.21.)
5. An operator algebra Let P(~) = L:a ba~a be a polynomial in n variables with constant coefficients and let P(D) be the associated differential polynomial, that is, the operator given by a
It follows from (1.19) that
(P(D)ff(~)
= P(27ri~)j(~).
Define the operator A to be the square root of the positive operator b.: (4.14) If P is a homogeneous polynomial of degree m, then
(4.15) where the operator T is defined by
.a
(Tff(~) = im~\~ j(~).
5. An operator algebra
81
The multiplier P(~)/I~lm is homogeneous of degree zero and its restriction to snl coincides with p(e), so it is a C= function. (In fact, it can be shown that T is a linear combination of compositions of Riesz transforms.) The operator T is not a singular integral of the type given in (4.1) since P(~) may not have zero average on snl. However, we can get such a singular integral by subtracting a constant, that is, a multiple of the identity operator. This particular operator T is a special case of the operators in the following result.
Theorem 4.13. If m E C=(JRn \ {O}) is a homogeneous function of degree 0, and Tm is the operator defined by (Tmff = mj, then there exist a E C and n E c=(snl) with zero average such that for any f E 5, Tmf
n(x')
= af +p.v. W
*f.
Since any homogeneous function of degree 0 is the sum of a constant and a homogeneous function of degree 0 with zero average on snl, Theorem 4.13 is an immediate consequence of the following lemma.
Lemma 4.14. Let m E C=(JR n \ {O}) be a homogeneous function of degree o with zero average on snl. Then there exists n E C= (snl) with zero average such that m(~) = p. v. n(x')/Ixl n . Proof. Since m is a tempered distribution,
m exists.
Hence,
(~:;) (~) = C~fm(O, where C is a constant. The function anm/axi is homogeneous of degree n, in C=(JR n \ {O}) and has zero average on snl. Furthermore,
anm
anm
a a = p. v. a + "" x'l!x'l!~ C a D 6, 2
2
lal:=;k
where 6 is the Dirac measure at the origin, since the difference between anm/a~i and p. v. anm/a~i is a distribution supported at the origin. If we take the Fourier transform of both sides of this equation we get
C~rm(~) =
(p.
v.
~:r;:) (~) + L 2
Ca (2rriO"·
lal:=;k
The left hand side and the first term on the righthand side are homogeneous distributions of degree 0, so the polynomial on the righthand side reduces to a constant. Thus the righthand side is a homogeneous function
82
4. Singular Integrals (I)
of degree 0 which is in coo(IRn \ {O}). Since this is valid for 1 < i < n, m coincides on IR n \ {O} with a homogeneous function of degree no We will denote its restriction to snl by O. To see that 0 has zero average on snl, fix a radial function ¢ E S which is supported on the annulus 1 < [xl < 2 and which is positive on the interior. Then
m(¢) = { O[ (x[2 ¢(x) dx = c ( O(u) da(u), J~n x Jsnl where c> 0; furthermore, since ¢ is radial and m is homogeneous,
m(¢) = m(¢) = c' (
}snl
Finally, to see that ference,
m is
m(u) da(u) = O.
identical to p. v. f2(x')j[x[n, consider their dif
A
O(x')
mp.v·W' which is supported at the origin. If we take the inverse Fourier transform of this difference we get a polynomial which must be constant since both m and (p. v: f2(x')j[x[nf are bounded. Further, this constant must be zero since both m and f2 have zero average on snl. 0
Theorem 4.15. The set A of operators defined by Theorem 4.13 is a commutative algebra. An element of A is invertibte if and only if m is never zero on snl . Proof. To see that A is an algebra, it is enough to note that given m2 with associated operators Tml and Tm2 , Tml 0 Tm2 = Tm1m2 ·
ml
and
Since the identity has the function 1 as its multiplier, Tm is invertible if and only if Tl/m E A, and 1jm E coo(IRn \ {O}) precisely when m(~) I 0 for any ~ E snl. 0
The operator T in (4.15) associated with the polynomial P is thus invertible if and only if P(~) is never zero on snl, that is, if P(D) is an elliptic operator. If the coefficients of P are real, then m must be even and Am = (_il)m/2. Thus the problem of solving P(D)u = f reduces to solving (_il)m/2u = T 1 f. (For more on this operator, see Section 7.7.)
6. Singular integrals with variable kernel
83
6. Singular integrals with variable kernel Let P(x, D) be a homogeneous differential polynomial with variable coefficients:
L
P(x, D) =
ba(x)Da.
[a[=m Since for
f
E
S,
we have that
P(x, D)f(x) =
r P(x, 27ri~)j(~)e27rix.f; d(
J'J1n
Further, with A as defined by (4.14), this has a representation analogous to (4.15):
P(x, D)f = T(Am f),
(4.16) where T is defined by
T f(x) =
(4.17)
fJ
r
fJ(x, ~)j(~)e27riX.f; d~,
J'J1n (
c)
=
P(x, i~)
1~lm.
X, ':,
The function fJ is homogeneous of degree 0 in the variable ~. If we substitute the definition of j(~) into (4.17) we get .
Tf(x) =
r J'J1
fJ(x,~)
n
r J'J1
f(y)e 27riy .f; dy e27rix .f; d~.
n
Thus formally, (4.18)
Tf(x) =
r K(x, x  y)f(y) dy,
J'J1n
where (4.19) In other words, for fixed x, K(x, .) is the inverse Fourier transform of fJ(x, .). By Theorem 4.13, for each x there exists a constant a(x) and a function O(x,·) E c=(snl) with zero average on snl such that
K(x, z) = a(x)5(z)
+ p. v.
O(x, z') Izln .
4. Singular Integrals (I)
84
This example is the motivation for studying singular integrals with variable kernel: (4.20)
= lim
T f (x)
£>0
1.
Iyl>£
D(x, y')
I In Y
f (x  y) dy.
We can again apply the method of rotations. Theorem 4.16. Let D(x, y) be a function which is homogeneous of degree o in y and such that:
(1) D(x, y) = D(x, y); (2) D*(u) = sup ID(x,u)1
Ll(snl).
E
x
Then the operator T given by (4.20) is bounded on LP(IRn ), 1 < p
< 00.
Proof. Because D is odd, we can argue as in the proof of Corollary 4.8 (where the kernel does not depend on x) to get
(4.21 )
Tf(x)
=
7r
r
2 }snl
D(x,u)Huf(x)da(u),
f E S.
Therefore,
ITf(x)1 <
7r
r
2 }snl
D*(u)IHuf(x)1 da(u)
and the desired result follows at once from Proposition 4.6.
o
Theorem 4.17. If in the statement of Theorem 4.16 we replace (2) by
(4.22) for some q, 1
s~p (fsn' 1!1(x, u)lq da(U)) q = Bq < 00 < q <00,
then T is bounded onLP(IRn ), q'
< p < 00.
Proof. Apply Holder's inequality with exponents q and q' to get
(4.23)
IT f(x)1 <
7r
2
Bq (
r IHuf(x)lq'da(u)) }snl
l/q'
If we raise this to the q'th power and integrate with respect to x we get that
If condition (4.22) holds for some value of q, then it holds for any smaller value, so we obtain the desired inequality for all p, q' < p < 00. 0
7. Notes and further results
85
When n is even there is a result analogous to Theorem 4.17 whose proof resembles that of Theorem 4.12. It can be found in the first article by A. P. Calderon and A. Zygmund cited in Section 7.l. Operators of the form (4.17) are a special class of pseudodifferential operators. For more information, see Chapter 5, Section 6.9.
7. Notes and further results 7.1. References.
The method of rotations was introduced in an article by A. P. Calderon and A. Zygmund (On singular integrals, Amer. J. Math. 78 (1956), 289309). The examples in Section 1 come from an earlier article by them (On the existence of certain singular integrals, Acta Math. 88 (1952), 85139) which will be discussed in Chapter 5. The proof of Theorem 4.12 for even kernels is taken from course notes by A. P. Calderon (Singular integrals and their applications to hyperbolic differential equations, University of Buenos Aires, 1960). It can also be found in the books by Zygmund [22] and Neri [13]. For a general overview of singular integrals and their applications, see the survey article by A. P. Calderon (Singular integrals, Bull. Amer. Math. Soc. 72 (1966),427465). 7.2. Spherical harmonics.
A solid harmonic is a homogeneous harmonic polynomial, and its restriction to sn1 is called a spherical harmonic. The spherical harmonics of different degrees are orthogonal with respect to the inner product on L2(sn1), and among those of a fixed degree there exists an orthogonal subcollection. Thus L2(sn1) has an orthogonal basis composed of spherical harmonics. Note that if n = 2, the basis of spherical harmonics for L2(Sl) is just {e ik (} , k E Z}. The functions Yk(x)e7TlxI2, where Yk is a solid harmonic of degree k, are eigenfunctions of the Fourier transform:
(This is referred to as the BochnerHecke formula.) The utility of spherical harmonics for studying singular integrals is suggested by the following formula: for k > 1,
4. Singular Integrals (I)
86
For a singular integral with 0 E L2(snl) and even, one can give a different proof of Theorem 4.12 using the decomposition of 0 into spherical harmonics. (See Stein and Weiss [18, Chapter 6].) The properties and uses of spherical harmonics can be found in the books by Stein [15], Stein and Weiss [18] and Neri [13]. Another discussion of spherical harmonics, as part of the general theory of harmonic functions, can be found in the book by S. Axler, P. Bourdon and W. Ramey (Harmonic Function Theory, SpringerVerlag, New York, 1992). 7.3. Operator algebras.
Besides the operator algebra studied in Section 5, Calderon and Zygmund considered others of the same type. (See Algebras of certain singular integral operators, Amer. J. Math. 78 (1956), 310320.) In particular, let Aq be the set of operators
Tf(x) = af(x)
+ lim E>O
1
O(y') IInf(x  y) dy,
lyl>E
Y
where a E C and 0 E Lq(snl). If we define
IITllq = lal + IIOIILQ(snl), then there exists a constant Bq which depends only on q such that
IITI T211q < BqllTlllqllT21lq. 0
The set Aq is a semisimple commutative Banach algebra. For more information on algebras of singular integrals, see the article by A. P. Calderon (Algebras of singular integral operators, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 1855, Amer. Math. Soc., Providence, 1967). 7.4. More on the method of rotations.
In the proof of Theorem 4.17, we could deduce that from inequality (4.23) provided that
(4.24)
(
In knl (
) l/p
P/ql
IHuf(x)lq' dO"(u)
)
liT flip < Gllfllp
dx
< Gllfll p·
In the proof of Theorem 4.17 we used that inequality (4.24) is immediate if p > q'. However, there are also values p < q' for which (4.24) is true, and this allows us to improve the theorem. If f is the characteristic function of the ball, then one can show that (4.24) cannot hold unless
n p
nl q'
<+1.
7. Notes and further results
87
It is conjectured that if 1 < p < 00 and p and q' satisfy this condition, then inequality (4.24) holds. It is also conjectured that (4.24) holds for this range of p and q' if Hu is replaced by the associated maximal operator, H~, or by the directional maximal function, Mu. The following results are known: when n = 2 the conjecture for Hu was proved by A. P. Calderon and A. Zygmund (On singular integrals with variable kernel, Appl. Anal. 7 (1978), 221238); when n > 3 they proved it for 1 < p < 2. M. Cowling and G. Mauceri (Inequalities for some maximal functions, J, Trans. Amer. Math. Soc. 287 (1985), 431455) showed that the same range holds for Mu. M. Christ, J. Duoandikoetxea and J. L. Rubio de Francia (Maximal operators related to the Radon transform and the Calder6nZygmund method of rotations, Duke Math. J. 53 (1986), 189209) showed that the conjecture is true for all these operators and for p in the range 1
1) .
n+ < p < max ( 2, 2
7.5. L log L results. As we noted at the end of Section 2, if !1 E L log L(sn1) is even then the Fourier transform of p. v. !1(x')/lxl n is bounded. Further, Theorem 4.12 can be extended to the case when the even part of n is in L log L(sn1). (See the first article by Calderon and Zygmund cited in Section 7.1.) These are the best possible results. In the same paper, Calderon and Zygmund noted that if ¢ is any function such that ¢( t) It log t + 0 as t + 00, then there exists an even function !1 such that ¢(!1) E L1(sn1) but the Fourier transform of p. v. !1(x')/lxl n is unbounded. Later, M. Weiss and A. Zygmund (An example in the theory of singular integrals, Studia Math. 26 (1965), 101111) showed that given any such ¢ there exists an even n, ¢(!1) E L1 (sn1 ), and a continuous function f E LP (I~n) for every p > 1, such that for almost every x,
E~O
2
!11 (y, f(x  y) dy = J1Y1>E Y
limsup.f
00.
7.6. Weak (1,1) inequalities. Proposition 4.6 does not extend to weak (1,1) inequalities because the Lorentz space L 1 ,= is not a normed space. (See Chapter 2, Section 8.3.) Hence, we cannot use the method of rotations to prove weak (1,1) inequalities for singular integrals of the type (4.1) or even for the rough maximal function MO, defined in (4.5). However, the following results are known:
88
4. Singular Integrals (I)
If n E LlogL(snl) then Mn is weak (1,1). For n = 2 this was proved by M. Christ (Weak type (1,1) bounds for rough operators, Ann. of Math. 128 (1988), 1942); for n > 2 this is due to M. Christ and J. L. Rubio de Francia (Weak type (1,1) bounds for rough operators, II, Invent. Math. 93 (1988),225237). It is unknown whether Mn is weak (1,1) if n E Ll(snl). If n E LlogL(snl) then the operator T in (4.1) is weak (1,1). This was proved by A. Seeger (Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95105). Partial results were obtained earlier by M. Christ and J. L. Rubio de Francia (in the above papers) and by S. Hoffman (Weak (1,1) boundedness of singular integrals with nonsmooth kernels, Proc. Amer. Math. Soc. 103 (1988), 260264). By the counterexamples given above this is the best possible result for T for arbitrary n. However, it is unknown whether T is weak (1,1) when n E Ll(snl) and is odd. P. Sjogren and F. Soria (Rough maximal operators and rough singular integral operators applied to integrable radial functions, Rev. Mat. Iberoamericana 13 (1997), 118) have shown that for arbitrary n E Ll(snl), T is weak (1,1) when restricted to radial functions. 7.7. Fractional integrals and Sobolev spaces.
If ¢ E S then
in Section 5 we defined the square root of the operator 
~
by
More generally we can define any fractional power of the Laplacian by
Comparing this to the Fourier transform of lxia, 0 < a < n (see (4.3)), we are led to define the socalled fractional integral operator Ia (also referred to as the Riesz potential) by .
1
Ia¢(x) = 
"'fa
LI ll~n
¢(y)
X 
I  a dy,
Yn
where
Then with equality in the sense of distributions,
7. Notes and further results
89
If we consider the behavior of Ia on LP, then homogeneity considerations show that the norm inequality
(4.25) can hold only if 1 q
(4.26)
1 p
a
n
In fact, this condition is also sufficient for 1
< p < n/ a.
Theorem 4.18. Let 0 < a < n, 1
I{x E JRn : IIaf(xll > .\}I < C
(II~ll) n/(na)
For a proof, see Stein [15, Chapter 5]. Another proof uses the following inequality due to L. Hedberg (On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505510):
Iaf(x) < Callfll~p/nMf(x)lap/n,
(4.27)
1
< n/a,
where M is the HardyLittlewood maximal function. To prove it decompose the integral defining Ia into two parts, one over {x : Ix  yl < A} and the other over {x : Ix  yl > A}; apply Proposition 2.7 to the first part and Holder's inequality to the second. Now choose A so that both terms are the same size. Theorem 4.18 then follows from (4.27) and from the boundedness ofM. Closely related to the fractional integral operator is the fractional maximal function,
Maf(x) = sup
I 1;_a /n JB{
r>O Br
If(x  y)1 dy,
0
< a < n.
T
Ma satisfies the same norm inequalities as Ia. The weak (1, n/(n  a)) inequality can be proved using covering lemma arguments (see Chapter 2, Section 8.6), and Ma is strong (n/a, (0): by Holder's inequality
IBrl;a/n
inr If(x  yll dy < (inc If(x  ylln/a dy
t
n
< IlfUn/a
The strong (p, q) inequalities then follow by interpolation. For any positive f it is easy to see that Maf(x) < Claf(x). The reverse inequality does not hold in general, but the two quantities are comparable In norm.
90
4. Singular Integrals (I)
Theorem 4.19. For 1 Ca,p such that
00
and 0
< a < n, there exists a constant
This result is due to B. Muckenhoupt and R. Wheeden (Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261274). Also see D. R. Adams, A note on Riesz potentials (Duke Math. J. 42 (1975), 765778) and D. R. Adams and L. Hedberg (Function Spaces and Potential Theory, SpringerVerlag, Berlin, 1996). An important application of Theorem 4.18 is in proving the Sobolev embedding theorem. For 1 < p < 00 and k > 1 an integer, define the Sobolev space L~(IRn) to be the space of all functions f such that f and all its derivatives (defined as distributions) up to order k are in LP. More precisely, let L~ be the closure of Cr;: under the norm
IIfIIL~ =
L
!IDa flip·
lal~k
Theorem 4.20. Fix 1 < P <
00
and k > 1 an integer. Then
(1) if p < nlk, then L~(JRn) C LT(JRn ) for all r, p 1/q = lip  kin; (2) if p = nlk, then L~(JRn) C LT(JRn ) for all r, p < r
< r < q, where <
00;
(3) if p > nlk and f E L~(JRn), then f differs from a continuous function on a set of measure o.
This result is due to S. L. Sobolev (On a theorem in functional analysis, Mat. Sb. 46 (1938), 471497; English translation in Amer. Math. Soc. Transl. ser. 2, 34 (1963), 3968). For all the results in this section, see Stein [15, Chapter 5], the book by Adams and Hedberg cited above, and the books by R. Adams (Sobolev Spaces, Academic Press, New York, 1975) and W. Ziemer (Weakly Differentiable Functions, SpringerVerlag, New York, 1989).
Chapter 5
Singular Integrals (II)
1. The Calder6nZygmund theorem In the previous chapter we used the Hilbert transform to study singular integrals. In this chapter we are going to consider singular integrals whose kernels have the same essential properties as the kernel of the Hilbert transform. This will let us generalize Theorem 3.2 to get the following result. Theorem 5.1 (Calder6nZygmund). Let K be a tempered distribution in ffi.n which coincides with a locally integrable function on ffi.n \ {O} and is such that
Ik(~)1 < A,
(5.1) (5.2)
1
Then for 1 < p
ixi>2iyi
<
IK(x  y)  K(x)1 dx < B,
y E ffi. n .
oo!
and
We will show that these inequalities are true for f E S, but they can be extended to arbitrary f E LP as we did for the Hilbert transform. Condition (5.2) is usually referred to as the Hormander condition; in practice it is often deduced from another, stronger, condition called the gradient condition.

91
5. Singular Integrals (II)
92
Proposition 5.2. The Hormander condition (5.2) holds if for every x =I 0
(5.3)
This follows from the mean value theorem; details are left to the reader. Proof of Theorem 5.1. Since this proof is (essentially) a repetition of the proof of Theorem 3.2, we will omit the details.
Let f E S and let Tf = K *f. From (5.1) it follows that IITfl12 < A11J112. It will suffice to prove that T is weak (1,1) since the strong (p,p) inequality, 1 < p < 2, follows by interpolation, and for p > 2 it follows by duality since the adjoint operator T* has kernel K*(x) = K( x) which also satisfies (5.1) and (5.2). To show that f is weak (1, 1), fix A > 0 and form the CalderonZygmund decomposition of f at height A. Then, as in Theorem 3.2, we can write f = g + b, where g E L2 and is bounded by 2n A and b is the sum of functions which are supported on disjoint cubes Qj and have zero average. The argument now proceeds as before, and the proof reduces to showing that
r
(5.4)
}~n\Q;
ITbj(x)1 dx < C
r Ibj(x)1 dx,
}Qi
where Qj is the cube with the same same center as Qj and whose sides are 2fo times longer. Denote their common center by Cj. Inequality (5.4) follows from the Hormander condition (5.2): since each bj has zero integral, if x tj Qj
Tbj(x)
=
r
}Qi
K(x  y)bj(y) dy
=
r [K(x 
}Qi
y)  K(x  cj)]bj(y) dy;
hence,
r
}~n\Q~J
ITbj(x)1 dx <
r Ibj(y)1 ( }~n\Q* r IK(x  y)  K(x  cj)1 dX) dy.
}QJ'
J
Since }R.n \
Qj c {x
E}R.n :
Ix  cjl
>
21Y  cjl},
the term in parentheses is bounded by B, the constant in (5.2).
D
We now consider singular integrals with a homogeneous kernel of degree n, such as we studied in the previous chapter: K(x) = Q(x')/Ixl n . What must we assume about Q for the Hormander condition (5.2) to hold? Clearly,
1. The Calder6nZygmund theorem
93
c
n E 1 (snl) is sufficient by Proposition 5.2. But much weaker conditions are sufficient. For example, define
woo(t)
=
sup{ln(Ul)  n(u2)1 : lUI  u21 < t, Ul, U2 E snl}.
Then we have the following result. Proposition 5.3. If n satisfies
1 1
(5.5)
woo(t) d
o
t
t<
00,
then the kernel n(x')/lxl n satisfies (5.2). Condition (5.5) is referred to as a Dinitype condition. Proof.
IK(x  y)  K(x)1
= n((x  y)')
n(x') Ixln
Ix  yin
< Ifl((x
I I fl(x') I + Ifl(x') y)') x_yn
1
Ix  yin
1 
Ixl n
Condition (5.5) implies that n is bounded, so on the set {Ixl > 21yl} the second term is bounded by Clyl/lxl n+1 . Thus its integral on this set is finite. On this set we also have that
I(x so
y)' 
x'i < 4 :~:,
1
1
In((x  y)')  n(x')1 d w oo (4lYl/lxl) d x< x Ixl>2lyl Ix  yin  Ixl>21YI (lxI/2)n
= 2nlsn11
r2 woo(t) dt
Jo
t
For similar results, see Section 6.2. From Proposition 5.3 and Corollary 4.5 we immediately get the following corollary to Theorem 5.1. Corollary 5.4. If n is a function defined on snl with zero integral and satisfying (5.5), then the operator
Tf(x) = p. v. is strong (p,p), 1 < p
n(y') J'rR lYF f(x n
< 00, and weak
(1,1).
y) dy
5. Singular Integrals (II)
94
Since (5.5) implies that n is bounded, we already had the strong (p, p) inequality via the method of rotations. Nevertheless, we now also have the weak (1,1) inequality. 2. Truncated integrals and the principal value In Theorem 5.1 we assumed that the operator was bounded on L2 (via the hypothesis that k E LOO). In this section we give conditions on K which imply this property and which make the associated operator bounded on LP, 1 < p < 00. Given K E Lfoc(IRn \ {O}), define KE,R(X) = K(X)X{E<\X\
Proposition 5.5. Let K E Lfoc(lRn \ {O}) be such that
( K (x) dx < A, Ja<\x\
(5.6)
(
(5.7)
Ja<\x\<2a
0
< a < b<
IK (x) I dx < B,
a
( IK(x  y)  K(x)1 dx < C, J\x\>2\y\
(5.8)
00;
> 0; y E JR n .
Then for all ~ E JR n , 1K:fi(~) I < C, where C is independent of E and R.
Condition (5.7) is equivalent to
1
(5.9)
IxIIK(x)1 dx < Bla,
a> O.
\x\
To see this, note that
and
{ IK(x)1 dx < ( EJ1K(x)1 dx < 2B'. Ja<\x\<2a J\x\<2a a Proof. If we fix ~ then whenever
K:;(~)
=
1 1
E
<
I~Il
< R,
K(x)e 27rix .t; dx
E<\x\
=
«\x\<\t;\l
=h+h.
K(x)e 27rix .t; dx
+ { J\t;\l <\x\
K(x)e 27rix .t; dx
2. Truncated integrals and the principal value
(If 1~11 <
95
or if 1~11 > R, then it suffices to consider just one of the two integrals.) We treat h first:
h
E
=
1
K(x) dx
+
E
1
K(x)(e27rix.~ 
1) dx,
E
so
Ihl <
1
K(x) dx
E
+ 27r1~ll
E
IxIIK(x)1 dx < C(A + B).
To evaluate h, let z = ~~1~12, so that exp(27riz·~) = 1; then if we make the change of variables x f+ x  z in h we get .
r
h =
JI~Il
K(x 
z)e27riX.~ dx.
Hence,
2h =
r
JI~Il
K(x)e27rix.~ dx 
r
JI~Il
K(x 
z)e27rix.~ dx;
this in turn implies that
21hl <
r
JI~Il
IK(x)  K(x  z)1 dx
r IK(x)1 dx + r IK(x)1 dx. J R ~ I~Il
+
J~ 1~Il
The first integral is bounded by C since 1~11 = 21zl. The other two integrals are bounded by 2B: since 1~11 < R, R+~I~I1 < 3(R~I~I1). D Corollary 5.6. If K satisfies the hypotheses of Proposition 5.5, then
IIKE,R
* flip < Cpllfllp,
1
00,
and
I{x
E}Rn :
IKE,R
C1
* f(x)1 > '\}I < >:llfI11'
with constants independent of E and R. When p = 2 this result follows immediately from Proposition 5.5; for the rest note that (5.8) implies the Hormander condition,
1
IKE,R(X  y)  KE,R(X) I dx < C, Ixl>2lyl with C independent of E and R. The details are left as an exercise for the reader.
96
5. Singular Integrals (II)
When the kernel is homogeneous, K(x) = Q(x')/Ixl n, conditions (5.6) and (5.7) are equivalent to Q E L1(sn1) and
hnl
Q(u) da(u)
=
O.
Since KE,R E L1, KE,R * f is well defined for f E LP. Ideally we could define the singular integral T by the pointwise limit
Tf(x) = lim KE,R * f(x), E+O R+= but this limit need not exist even if f E S. In this case we have convergence as R + 00, so the problem reduces to determining when the principal value distribution of K, p. v. K(¢)
=
lim ( K(x)¢(x) dx, E+O JIxI>E
¢
E
S,
exists.
Proposition 5.7. Given a function K which satisfies condition (5.7)7 the tempered distribution p. v. K exists if and only if lim! K(x)dx E+O E
exists. Proof. Suppose that the tempered distribution exists. If we fix ¢ E S which is identically 1 on B(O, 1), then p. v. K(¢)
=
lim! K(x) dx E+O E
+ (
Jxl.>1
K(x)¢(x) dx.
1
The second integral exists since
(
Jxl>1
IK(x)¢(x)1 dx < Illxl¢ll= f2 k / , k=O
1
IK(x)1 dx < 2Blllxl¢II=·
2k
Therefore, the limit of the first integral must also exist. Conversely, suppose the limit exists; denote it by L. Then p. v. K(¢)
= ¢(O)L +
1
K(x)(¢(x)  ¢(O)) dx +
Ixl
(
Jxl<1 1
1
K(x)¢(x) dx.
Ixl>l The first integral exists since
IK(x)II¢(x)  ¢(O)I dx < IIV¢II= { IxIIK(x)1 dx < 2BIIV¢II=· Jxl<1 1
o
2. Truncated integrals and the principal value
97
Corollary 5.8. If we add to the hypotheses of Proposition 5.5 the assumption that
lim! K(x) dx E
E>O
exists, then Tf(x)
= lim E>O
1
lyl>E
K(y)f(x  y) dy
can be extended to an operator which is bounded on LP, 1 weak (1,1).
00,
and is
Example 5.9 (An application to K(x) = Ixlnit). The function Ixl n  it is locally integrable on }Rn \ {O} and satisfies the hypotheses of Proposition 5.5. In fact,
1
dx a
=
{
IsnII bit  .a it < ~ IsnIl ~t  t ' dx
=
Isn I ll og 2
Ja
'
In + itl IDK()I v x < Ixl n+I ' Hence,
IIKE,R
* flip
< Cpllfllp with Cp independent of E and R.
However, the limit of the truncated integrals does not exist almost everywhere since
1
dx. E
=
IsnI11  E it it '
and the limit of E it as E + 0 does not exist. The limit does exist if we choose an appropriate subsequence {Ek}, for example, Ek = e 27rk / t . If we take this sequence then
.
hm KEbR
k,R>=
* f(x) =
1.
f(xy)f(x) IY In+it dy lyl<1
+
1.
f(xy) I In+it dy, Iyl>l Y
and this defines an operator which is strong (p,p), 1 < p < 00, and weak (1,1). This operator is the convolution of f with the tempered distribution which we will refer to as p. v.lxl n  it , even though it depends on our choice of the sequence {Ek}: 1 p. v. Ixln+it
(¢)
=
1
Ixl
¢(x)  ¢(O) Ixln+it
dx
+
1
¢(x)
Ixl>1 Ixln+it dx.
5. Singular Integrals (II)
98
Despite its appearance this distribution is not homogeneous of degree n  it (see Definition 4.2); that is, it does not satisfy 1 ( ) \nit 1 (A.) p. v. Ixln+it ¢)... = /\ p. v. Ixln+it 'f' ,
where ¢)...(x) = An¢(AIX). This can be shown by a straightforward computation, but here we are going to give an indirect argument which also yields its Fourier transform. Given z such that Re z < n, the function Ixl z is locally integrable and homogeneous of degree z. It defines a tempered distribution which is also homogeneous of degree z:
We can rewrite this as _1 (¢) Ixl z
¢(x)  ¢(O) dx + ¢(O) {
= { J1xl<1
¢(x) /(0) dx +
= { J1xl<1
~
+ {
J1xl<1 Ixl z
Ixl z
{
J1xl>1 Ixl z
¢(x} dx + IsnIl ¢(O);
J1xl>1 Ixl
Ixl
¢(x) dx
n  z
this expression makes sense for Re z < n+ 1 except if z = n. This distribution is homogeneous of degree and if we let z = n + it, we see that it differs from the one we call p. v.lxl n  it by a multiple of the Dirac delta. (This is homogeneous of degree n, which implies that p. v.lxlnit cannot be homogeneous.) This new distribution is thus the unique distribution that is homogeneous of degree n  it and which coincides away from the origin with the locally integrable function Ixl n it . Given the Fourier transform of Ixl z (see (4.'3)),
z,
(
1 ) ~
.r;F
(~) =
n
z 7r 
2
r (n1 r (;) I~Iz+n' Z )
if we let z = n + it we get the Fourier transform of the homogeneous distribution, which in turn gives us the Fourier transform of p. v.lxl n  it :
1) (t) _ ( Ixln+it ~
p. v.
<:,

2
(it)
~+it rr (ntit) I~ lit + ~IsnIl 7r it .
3. Generalized CalderonZygmund operators Up to this point the operators we have been studying could be written as convolutions with tempered distributions. One practical advantage of this hypothesis is that we could use the Fourier transform to deduce the £2 boundedness of the operator. If we assume this then the boundedness on
3. Generalized Calder6nZygmund operators
99
the remaining LP spaces only depends on the Hormander condition, which can be adapted to operators which are not convolution operators. Let.6. be the diagonal offfi.n x ffi.n: .6. = {(x,x): x E ffi.n}. By a proof nearly identical to that of Theorem 5.1 we can show the following. Theorem 5.10. Let T be a bounded operator on L2(ffi.n)) and let K be a function on ffi.n X ffi.n \ ~ such that if f E L2(ffi.n) has compact support then
=
Tf(x)
r
K(x, y)f(y) dy, Jrr:r.n Further) suppose that K also satisfies
x
~ supp(f).
r IK(x, y)  K(x, z)1 dx < C, r IK(x, y)  K(w, y)1 dy < J1xyl>2Ixwl .
(5.10)
.
J1xyl>2Iyzl
(5.11)
C.
Then T is weak (1,1) and strong (p,p)) 1 < p <
00.
The given representation holds only for functions in L2 of compact support, but this suffices for the proof since it is only applied to the functions bj gotten from the Calder6nZygmund decomposition. condition (5.10) is used to prove the weak (1,1) inequality for T, and condition (5.11) is used to prove the same inequality for its adjoint. Following the terminology of Coifman and Meyer [2], we say that K : ffi.n X ffi.n \ ~ ~ C is a standard kernel if there exists <5 > 0 such that C
(5.12)
IK(x,y)1 < 1x_yn I'
y) 
(5.13)
IK(x,
(5.14)
' y) IK (x,
K(x, z)1 < K( w, y )1
c I;yyil:,
b < C xIxwl _ y n+b 1
1
if
Ix  yl > 21Y  zI,
l'f
1X  Y1> 21Ix 
w 1.
Standard kernels dearly satisfy the Hormander conditions (5.10) and (5.11). The following are examples of operators which have this type of kernel.
(1) The Cauchy integral along a Lipschitz curve. Let A be a Lipschitz function on ffi. (i.e. A' = a E L oo ) and let r = (t, A(t)) be a plane curve. With this parameterization we can regard any function f defined on function of t and conversely. Given f E S(ffi.), the Cauchy integral
roo
f(t)(l + ia(t)) dt 27ri J00 t + iA(t)  z defines an analytic function in the open set
Cr f(z) = _1
D+ = {z = x
+ iy E C : y >
A(x)}.
r
as a
5. Singular Integrals (II)
100
Its boundary values on
r, lim Cr f(x
E+O
are given by
~
[f(X)
2
+ i(A(x) + E)),
1
+ ~ lim
f(t)~l + ia(t)). dtj. Ixtl>EXt+'l,(A(x)A(t))
7rE+O
This leads us to consider the operator
Tf(x) = lim E+O
1
IxYI>E
f(y) dy, X  Y + i(A(x)  A(y))
whose kernel, 1
(5.15)
K(x, y) = x  y + i(A(x)  A(y)) ,
satisfies (5.13) and (5.14) with 8
= 1.
(2) Calderon commutators. If as a geometric series:
K(x,y)
= _1_
IIA'lloo
then we expand the kernel (5.15)
~ (iA(x) 
A(y))k xy
xyD k=O
It is, therefore, natural to consider the following operators: given a Lipschitz function A on IR. and an integer k > 0, define
Tkf(x) = lim .
E+O
1
(A(X)A(y))k f(y) dy.
IxYI>E
X 
Y
x  y
The associated kernels,
Kk(x,y) = (A(X) A(Y))~_l_, xy xy are also standard kernels with 8 = 1. Definition 5.11. An operator T is a (generalized) Calder6nZygmund operator if
(1) T is bounded on L2(IR.n);
(2) there exists a standard kernel K such that for support,
T f(x) =
r K(x, y)f(y) dy,
JlRn
f
E
L2 with compact
x tf supp(f).
Theorem 5.10 implies that a Calder6nZygmund operator is bounded on LP,l < p < 00, and is weak (1, 1). Hence, given an operator with a standard kernel, the problem reduces to showing that it is bounded on L2. We will return to this question in Chapter 9.
4. Calder6nZygmund singular integrals
101
4. Calder6nZygmund singular integrals The examples in the previous section suggest that for any Calder6nZygmund operator, T f(x)
=
lim ( K(x, y)f(y) dy, J1xyl>E
EO
at least for f E S. However, this is not necessarily the case. From (5.12) we can deduce that TEf(x) =
K(x, y)f(y) dy
(
J1xyl>E makes sense for f E S(Il~n). Nevertheless, the limit as E + 0 need not exist or may exist and be different from T f (x). An example of the first is the operator we defined in Section 2 as convolution with p. v.lxl n  it , whose kernel, K(x, y) = Ix  ylnit, is standard. (Recall that this operator is strong (2,2) and so a Calder6nZygmund operator.) By an argument identical to that in Proposition 5.7 we can prove the following result. Proposition 5.12. The limit
(5.16) exists almost everywhere for f E
C~
if and only if
lim ( K(x, y) dy JE
EO
exists almost everywhere.
The existence of the limit (5.16) does not imply that it is equal to Tf(x): for example, consider the identity operator I. This is a Calder6nZygmund operator associated with the kernel K(x,y) = Oclearly, If(x) = 0 if x rfsupp(f)but (5.16) equals O. Furthermore, this example shows us that an operator is not characterized by its kernel since the zero operator also has the zero kernel. In general, so does any pointwise multiplication operator Tf(x) = a(x)f(x), a E Loo. However, these are the only ones; this is shown by the following result whose proof is left to the reader. Proposition 5.13. If two Calder6nZygmund operators are associated with the same kernel) then their difference is a pointwise multiplication operator.
It is important to assume that a Calder6nZygmund operator is bounded on L2 since without this or a similar hypothesis, Proposition 5.13 would be
5. Singular Integrals (II)
102
false. For example, the derivative is an operator with kernel 0 (f' (x) x FJ supp(f)) but it is not a pointwise multiplication operator.
= 0 if
A Calder6nZygmund singular integral is a Calder6nZygmund operator that satisfies
Tf(x)
(5.17)
=
lim TEf(x).
E~O
To determine when this equality holds for f E LP(n~n), we will proceed as we did for the Hilbert transform and examine the maximal operator
T* f(x)
=
sup ITEf(x)l. E>O
By Theorem 2.2, if T* is weak (p,p) then the set
{f E LP : lim TEf(x) exists a.e.} E~O
is closed in LP. Hence, if (5.17) holds for a dense subset, say it holds for any f E LP.
f
E C~, then
Theorem 5.14. If T is a Calder,'6nZygmund operator then T* is strong (p,p), 1 < p < 00, and weak (1,1). The proof of Theorem 5.14 depends on the following result which is analogous to Cotlar's inequality (Lemma 3.5).
Lemma 5.15. If T is a Calder6nZygmund operator then for any lJ < 1, and for any f E C~,
(5.18)
T* f(x) < C v (M(IT fIV)(x)l/V
+ M f(x))
lJ,
0
<
.
To prove this we need the following result due to Kolmogorov.
Lemma 5.16. Given an operator S which is weak (1,1), lJ, 0 < lJ < 1, and a set E of finite measure, there exists a constant C depending only on lJ such that
Proof. By (2.1) and the weak (1,1) inequality,
LISf(x)IV dx = v 1000 AVll{x .< v =
v
E
E : ISf(x)1 >
10= )."1 min (lEI, ~ Ilf111)
l
C11f"d'E'
o
A}I dA d)'
x/lIEI dA + v 100 CIIfildlEI
CA v 
2
11f1h dA.
4. Calder6nZygmund singular integrals
103
The desired inequality follows from this immediately.
D
Proof of Lemma 5.15. We will prove that
with C independent of E. Our argument will be translation invariant and so actually yields (5.18). Fix E > 0; let Q = B(O, E/2) and 2Q and 12 = f  h· Then
T 12(0) =
!
= B(O, E), and define h =
fX2Q
K(O, y)f(y) dy = TEf(O).
IYI>E If z E Q then, since K is a standard kernel,
JTh(z)  Th(O)1
=!
lyl>E
< Cl z l8

< 
<
ci
(K(z,y)  K(O,y))f(y) dy
!
f
If(y)1 dy
IYI>E lyln+8
r
k=O J2 k E
lJ(y) 1 dy lyln+8
efT" (2k1 )n!lyl<2 + k=O
E
k
If(y)1 dy 1E
< C8 M f(0). It follows from this inequality that
ITEf(O)1 < CMf(O)
(5.19) If TEf(O)
0< >.
+ ITf(z)1 + ITh(z)l·
then there is nothing to prove. If not, fix < ITEf(O)1 and let = 0
Q1 = {z E Q : ITf(z)1 > >./3}, Q2 = {z E Q : ITh(z)1 > >./3}, and if CM f(O) <
>'/3, if CM f(O) > >./3.
Then Q = Q1 U Q2 U Q3, so IQI < IQ11
IQ11 < }
+ IQ21 + IQ31.
However,
fa ITf(z)1 dz < }IQIM(Tf)(O),
>. such that
5. Singular Integrals (II)
104
and by the weak (1,1) inequality for T,
IQ21 = If Q3
I{z
E
Q : ITh(z)1 >
3C ( 3 C TIQIMf(O).
A/3}1 < T i Q If(z)1 dz <
= Q then A < 3CM f(O); if Q3 = 0 then
IQI ~ IQll + IQ21 < 3~IQI(M(TJ)(0) + Mf(O)). Hence, in every case we have that
A < C (M(TJ) (0)
+ M f(O)).
This is true for any A < ITEf(O)1 and so (5.18) holds when v
=
l.
If 0 < v < 1, then it follows from (5.19) that
ITEf(O)I V < CMf(ot
+ ITf(z)IV + ITh(z)l v . If we integrate in z over Q, divide by IQI and raise to the power l/v, we get IT,1{O)I" < C ( Mf{O)
+ M{ITfl"){O)1/" +
(I~I fa ITh{z)l" dZ) 1/") .
By Lemma 5.16, (
1
(
lQJ i Q ITh(z)I Vdz
) l/v
< CIQIlllhih < CMf(O),
and this completes the proof.
D
Proof of Theorem 5.14. Inequality (5.18) with v = 1 immediately implies that T* is strong (p, p) since both T and Mare.
To show that T* is weak (1, 1) we could argue as we did in the proof of Theorem 3.4; instead we will give a different proof which uses (5.18) with v < l. From this inequality we have that
I{x
E}Rn :
T* f(x) >
A}I
E}Rn :
Mf(x) > A/2C}1
+ I{x E}Rn : M(ITfIV)(x)l/V > A/2C}I, and the first term on the righthand side satisfies the desired estimate. As for the second term, by Lemma 2.12
where Md is the dyadic maximal operator (2.9). Let E = {x E Md(ITfIV)(x) > AV}; then E has finite measure if f E C~. But then
lEI <
:v LITf(Y)I Vdy;
}Rn :
5. A vectorvalued extension
105
by the proof of Theorem 2.10 it suffices to take the integral over E instead of all of ]Rn. Therefore, by Lemma 5.16 lEI < C,\vIEI 1  v llfllr· The desired inequality now follows if we replace
,\1/
o
by 4 n ,\1/.
5. A vectorvalued extension Let B be a separable Banach space. A function F from]Rn to B is (strongly) measurable if for each b' E B* (the dual of B) the map x r+ (F(x), b') is measurable. If F is measurable then it follows that the scalar function x r+ IIF(x)IIB is also measurable. Therefore, we can define LP(B) as the space of (equivalence classes of) measurable functions from ]Rn to B such that
IIFIILP(B) =
(IF!' IIF(x)ll~ dX) IIp <
and Ilflloo = sup{IIF(x)IIB : x E Banach space.
00,
1
00,
The space LP(B), 1 < p <
]Rn}.
00,
is a
If f is a scalar function in LP and b E B, define the function f . b from ]Rn to B by (f· b)(x) = f(x)b. This function is in V(B) and its norm is IlfllpllbllB. The subspace of LP(B) consisting of finite linear combinations of functions of this type, denoted by LP ® B, is dense if 1 < P < 00.
Given F B given by
=
L:j fJ . bj
L
E
L1 ® B, define its integral to be the element of
F(x) tk
=
L
(in !j(X) dX) bj
J
The map F r+ f F(x) dx extends to L1(B) by continuity. For a function F E L1(B), this integral is the unique element of B that satisfies
(in F(x) dx, b'J L(F(x), b') dx =
for all b' E B*. If F E LP(B) and GELpI (B*), then (F, G)(x) = (F(x), G(x)) is integrable; furthermore,
IIGlid (B» = sup {
.L
From this we see that Lpl (B*) but is, for example, if 1 < p <
(F(x), G(x)) dx : IIFIILP(B) < 1 } .
c (V(B))*. Equality is not true in general, 00
and B is reflexive.
Let A and B be Banach spaces and let £(A, B) be the space of bounded linear operators from A to B. Suppose that K is a function defined on
5. Singular Integrals (II)
106
~n X ~n \
Ll which takes values in £(A, B) and T is an operator which has K as its associated kernel: if f E DX)(A) and has compact support, then
T f(x)
=
r K(x, y) . f(y) dy,
x tJ. supp(f)·
J~n
For such operators there is a vectorial analogue of Theorem 5.10. Theorem 5.17. Let T be a bounded operator from LT(A) to LT(B) for some r, 1 < r < 00, with associated kernel K. If K satisfies
(5.20) (5.21)
r
IIK(x, y)  K(x, z) II£(A,B) dx :::; C,
1
IIK(x, y)  K(w, y)II£(A,B) dy < C,
J1xyl>2Iyzl Ixyl>2Ixwl
then Tis bounded from £p(A) to LP(B), 1 < p <
00,
and is weak (1, I), that
'is,
I{x
E
~n:
IITf(x)IIB > A}I <
~llfll£l(A).
The proof of Theorem 5.17 initially follows the outline of the scalar result: inequality (5.20) together with the boundedness from LT(A) to LT(B) yields the weak (1,1) inequality; then the Marcinkiewicz interpolation theorem (which can easily be shown to be true in these spaces) shows that T is bounded for 1 < p < r. To prove that T is bounded for r < p < 00 we must pass to the adjoint operator. However, this presents a problem since, as we noted above, Lpf (A*) need not be equal to LP(A)*. When A is reflexive (and it will always be so in our applications in Chapter 8) they are equ<;1l, and in that case it is enough to note that the kernel associated with the adjoint operator T* is K(x, y) = K*(y, x) E £(B*, A*). Inequality (5.21) for K is equivalent to (5.20) for K, so by repeating the above argument we get that T* is bounded for 1 < p < r'o When Lpf (A*) 0:/= LP(A)* we must first consider the finite dimensional subspaces of A. Given such a subspace A o, let To : LT(Ao) ~ LT(B) be the restriction of T to functions with values in Ao. The kernel associated with To isKo E £(Ao, B), the restriction of K to Ao· Since IIKol1 < IIKII, inequalities (5.20) and (5.21) hold for Ko with constants independent of the subspace Ao. Therefore, arguing as before,
T; : Lq(B*)
~
Lq(Ao),
1 < q < r',
with a constant independent of Ao. So by duality, To is bounded from LP(A o) to LP(B), r < p < 00, with a constant independent of Ao. That it is bounded on all of LP (A) now follows since LP 0 A is dense in LP (A) .
6. Notes and further results
107
We leave it to the reader to fill in the details of the outlined proof of Theorem 5.17 by following the scalar model.
6. Notes and further results 6.1. References.
The CalderonZygmund decomposition and its application to proving the weak (1,1) inequalities for singular integrals appeared in the classic article by both authors (On the existence of certain singular integrals, Acta Math. 88 (1952), 85139). In this article they used the gradient condition (5.3); L. Hormander first observed that (5.2) is sufficient (Estimates for translation invariant operators in £P spaces, Acta Math. 104 (1960), 93139). Also see Stein [15]. Proposition 5.5 is due to A. Benedek, A. P. Calderon and R. Panzone (Convolution operators with Banachspace valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356365). The tempered distribution p. v.lxl n it and its generalizations were first considered by B. Muckenhoupt (On certain singular integrals, Pacific J. Math. 10 (1960), 239261). For more on generalized CalderonZygmund operators, consult Coifman and Meyer [2], Journe [8] and Stein [17]. The Cauchy integral has a long history: see, for example, the book by N. T. lMushkelishvili (Singular Integral Equations, P. Noordhoff, Groningen, 1953). The associated commutators were first studied by A. P. Calderon (Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 10921099). Commutators are discussed again in Chapter 9. For more on vectorvalued singular integrals see GarciaCuerva and Rubio de Francia [6, Chapter 5], the article by J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea (Calder6nZygmund theory for operatorvalued kernels, Adv. in Math. 62 (1986), 748) or the monograph by J. L. Torrea (Integrales Singulares Vectoriales, Notas de Algebra y Analisis 12, Univ. Nacional del Sur, Bahia Blanca, 1984). The survey article by C. Fefferman (Recent progress in classical Fourier analysis, Proceedings of the I.C.M. (Vancouver, B.C., 1974), Vol. 1, pp. 95118, Canad. Math. Congress, Montreal, 1975) contains a succinct discussion of singular integrals and their connection to many other problems in analysis.
6.2. A more general Dinitype condition. The Hormander condition (5.2) holds for kernels of the form K(x) n(x')/Ixln under weaker hypotheses than those in Proposition 5.3. Given P E O(n), let Ilpll
= sup{lu  pul : u
E
snl}.
5. Singular Integrals (II)
108
If we define W1(t)
=
r
sup
Ilpll~t } snl
1r2(pu)  r2(u)1 dO"(u),
then the Dinitype condition
1
1 W1(t) d
o
  t
implies that (5.2) holds for K. This condition also implies that r2 E L log+ L. This result is due to A. P. Calderon, M. Weiss and A. Zygmund (On the existence of singular integrals, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 5673, Amer. Math. Soc., Providence, 1967). Later, Calderon and Zygmund proved that this condition is also necessary for the Hormander condition to hold (A note on singular integrals, Studia Math. 65 (1979), 7787). 6.3. N onisotropic dilations. The Euclidean norm
I . I on ]Rn is associated with a family of dilations 8t (x)
in the sense that
=
(tX1,'"
18t (x)1 = tlxl, t> 0,
,txn )
x E ]Rn.
Given a1,' .. ,an > 0, we can define a family of dilations that act on each variable differently: 8t (x) = (tal Xl, . .. ,tanxn ).
II . II such that 118t (x) II = tllxll Ilzll = 1. Define II . II as follows: given
Then for each 8t there exists a quasinorm
and if z E sn1 then x E ]Rn, there exists a unique x' E sn1 such that 8t (x ' ) = x for some t > O. Let Ilxll = t. This is not an actual norm since the triangle inequality is only satisfied up to a constant L > 1:
Ilx + yll < L(llxll + Ilyll)· Given]Rn with this quasinorm we can get results analogous to Theorem 5.10 by considering conditions like
r
hxYII~21Iyzll
IK(x, y)  K(x, z)1 dx
and the symmetric condition gotten by changing the order of the variables. This kind of singular integral appears primarily in connection with the parabolic operators introduced by E. Fabes and N. Riviere (Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 1938). Also see C. Sadosky ( On some properties of a class of singular integrals, Studia Math.
6. Notes and further results
109
27 (1966), 7386) and M. de Guzman (Singular integral operators with generalized homogeneity, Rev. Acad. Ciencias Madrid 64 (1970), 77137). In Chapter 8, Section 7, we will use the method of rotations to study the kernels associated with this structure.
6.4. Spaces of homogeneous type. Singular integrals can be studied in very general spaces. R. Coifman and G. Weiss considered this question in their book Analyse harmonique nonconmutative sur certains espaces homogenes (Lecture Notes in Math. 242, SpringerVerlag, Berlin, 1971). For a discussion in a slightly less general setting, see Stein [17, Chapter 1]. Let X be a space with pseudometric p, that is, a function p : X x X [0,00) such that
= 0 if and (2) p(x, y) = p(y, x); (1) p(x, y)
(3) there exists L
only if x
~
= y;
> 0 such that p(x, z) < L(p(x, y) + p(y, z)).
A space of homogeneous type is a topological space X with a pseudometric p such that the balls B(x, r) form a basis of open neighborhoods of X, and there exists a positive integer N such that for any x E X and r > 0, the ball B(x, r) contains at most N points Xi such that p(Xi' Xj) > r /2. The second property holds if there exists a Borel measure fJ on X such that
0< fJ(B(x,r)) < AfJ(B(x,r/2)) < 00. (Such a measure fJ is called a doubling measure. We first discussed doubling measures in Chapter 2, Section 8.6.) If X is a space of homogeneous type, then given a kernel K(x, y) E L2(X X X, dfJ 0 dfJ) we can define the operator
Tf(x) If
=
Ix
IITfllq < Cqllfllq for some q > 1,
(5.22)
(
K(x, y)f(y) dfJ(y). and if for any y, z E X,
IK(x, y)  K(x, z)1 dfJ(x) < C,
} p(x,y)?2Lp(y,z)
then T is strong (p,p), 1 < p
< q, and weak (1,1).
Very recently, in connection with the study of the Cauchy integral, the question has arisen of extending the theory of singular integrals to nonhomogeneous spaces: topological spaces X with a pseudometric p and a measure fJ which is not doubling. F. Nazarov, S. Treil and A. Volberg (W~ak type estimates and Cotlar inequalities for CalderonZygmund operators on nonhomogeneous spaces, Int. Math. Res. Let. 9 (1998), 463487) have shown
5. Singular Integrals (II)
110
that the above result remains true if: p is a metric; there exists a constant v > 0 such that for all x E X and r > 0, p,(B(x, r))
p(y, z)O IK(x,y)  K(x,z)1 < C p(x,y)v+o'
if p(x,y) ~ 2p(y,z).
6.5. The size of constants.
In the proof of Theorem 5.1, if we form the Calder6nZygmund decomposition at height AlA, then the weak (1,1) constant becomes Cn(A+B), and by interpolation the strong (p, p) constant is dominated by C~(A
+ B)p2
1
pl
00.
In general, this is the best possible constant asymptotically as p ~ 1 or p ~ 00. To see this, suppose the kernel K satisfies the gradient condition with constant C (5.3) and also satisfies c
IK(x)1 > Ixl n '
Ixl > O.
Let B = B(O, 1) and let B* = B(O, R), where R is such that C / R Fix f = IBIljpXB; then Ilfllp = 1. Further, for x tJ. B*,
ITf(x)1 > IBllljp IBIljp
>
clBllljp


Ix In
> 
clBllljp 2 Ixl n .
< c/2.
L
IK(x  y)  K(x)1 dy
CIBll+ljnl jp ,,=
Ixl n+l
It follows from this that for p small, IITfllp = O((p_1)1). If we apply the same argument to the adjoint operator, then by duality we get that for p large, IITfllp = O(p).
This proof can easily be adapted to the case when K is the kernel of one of the Riesz transforms, and a similar argument holds for Calder6nZygmund operators. (See Stein [17, p. 42].) 6.6. More on truncated integrals.
Given a Calder6nZygmund singular integral T, that is, a CalderonZygmund operator that satisfies (5.17), it is natural to ask when TEl converges to T f in LP. If 1 < p < 00, this is immediate: if f E LP then by Theorem 5.14, T* f E LP, so convergence follows from the dominated convergence theorem. This argument fails when p = 1, but nevertheless it can be shown that if f, T fELl, then TEf ~ T f in Ll. This result is due to
6. Notes and further results
111
A. Calderon and O. Capri (On the convergence in £1 of singular integrals, Studia Math. 78 (1984),321327). 6.7. Vectorvalued singular integrals and maximal functions. The vectorvalued extension of the theory of singular integral operators in Section 5 has wide application, since many of the operators in harmonic analysis can be viewed as vectorvalued singular integrals. This approach is due to Rubio de Francia, Ruiz and Torrea (see the reference given in Section 6.1). Here we will show that the HardyLittlewood maximal function can be treated as a vectorvalued singular integral. We first extend Theorem 5.17 to include the case when T is a bounded operator from LOO(A) to LOO(B). The proof is essentially the same. Fix f, form the CalderonZygmund decomposition of IlfilA at height >., and decompose f as g+b, where IIg(x)IIA < 2n>. a.e. Then, since T is bounded from LOO(A) to LOO(B), for some constant /3 > 0,
{x ElRn
:
IITf(x)liB > /3>'} C {x E lRn
:
IITb(x)liB > >.}.
The proof then continues as before. Now fix a nonnegative function ¢ E S such that ¢ Define the maximal operator
MqJ(x)
=
sup I¢t t>O
> 1 on
B(O,l).
* f(x)l,
where ¢t(x) = tn¢(x/t). Since ¢ E S, it suffices to take the supremum over rational t. By our choice of ¢, M f(x) < M¢>(lfl)(x). (The reverse inequality was proved in Proposition 2.7.) Thus, to prove norm inequalities for M it will suffice to prove them for M¢>. Let A = lR and B = ,eOO(Q+), and define the vectorvalued function K(x) = {¢t(X)}tEQ+' Then K takes values in £(A,B) and we can define an operator T to be convolution with K. Since
. IITf(x)liB = M¢>f(x), it follows that T : LOO(A)
+
LOO(B) is bounded. Further, since ¢
ely I
sup I¢t(x  y)  ¢t(x)1 < I In+1' t>O
x
E
S,
Ixl > 21YI > 0,
which implies the Hormander condition (5.20) for K. Hence, by Theorem 5.17 (as extended above), T : £P(A) + LP(B); equivalently, M¢> is bounded on LP. The boundedness of M¢> was originally proved by F. Zo (A note on approximation of the identity, Studia Math. 55 (1976), 111122).
5. Singular Integrals (II)
112
6.8. Strongly singular integrals.
In Theorem 5.1, given an arbitrary tempered distribution with bounded Fourier transform, the Hormander condition (5.2) seems to be the weakest hypothesis possible to get LP boundedness. It is of interest, however, to see how this hypothesis can be weakened if stronger conditions are assumed on the Fourier transform. The prototypical operator motivating this is the convolution operator Ta ,b = Ka ,b * f, 0 < a < 1 and b > 0, where ~
eil~la
Ka,b(e) = ¢(e)W' and ¢ is a Coo function which vanishes near the origin and is identically equal to one outside of some compact set. This integral arises in the study of the LP convergence of multiple Fourier series (cf. Chapter 1, Section 10.2). Clearly,
A IKa,b(e) I ::; (1 + lel)b' A
and it can be shown that for x close to the origin, B
IKa,b(X)1 ~ Ixln+8'
where 8 = (naj2b)j(1a). Hence, this kernel does not satisfy the gradient condition (5.3). (It also fails to satisfy the Hormander condition; this fact follows from the next result.) Howeve.r, these two growth conditions are sufficient to determine the behavior of Ta ,b on LP. Theorem 5.18. The operator Ta ,b is bounded on LP if
(5.23)
1
1
2]; <
(bjn)(nj2 + 8) b+8'
8'
naj2  b Ia'
and is unbounded if the reverse inequality holds. If equality holds in (5.23), then Ta,b maps LP into the Lorentz space LP'P' .
The LP boundedness of Ta,b was first shown by I. Hirschmann (On multiplier transformations, Duke Math. J. 25 (1959), 221242) when n = 1 and by S. Wainger (Special trigonometric series in k dimensions, Mem. Amer. Math. Soc. 59 (1965)) in higher dimensions. Also see the survey article by E. M. Stein (Singular integrals, harmonic functions and differentiability properties of functions of several variables, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 316335, Amer. Math. Soc., Providence, 1967). The endpoint inequalities were proved by C. Fefferman (Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 936). He showed that they are consequences of the following general result, which
6. Notes and further results
113
further illustrates the delicate relationship between the conditions on K and/(. Theorem 5.19. Let K be a tempered distribution on ~n with compact support which coincides with a locally integrable function away from the origin. Suppose there exists 0 < < 1, such that
e,
e
A IK(e)1 < (1 + lel)nB/2' ~
r
J xl>2Iylla
eE ~n,
IK(x  y)  K(x)1 dx < B,
Iyl < 1.
1
Then the operator T f = K a weak (1,1) inequality.
*f
is bounded on LP, 1 < p <
00,
and satisfies
This result was later generalized by P. Sjolin (LP estimates for strongly singular convolution operators in ~n, Ark. Mat. 14 (1976), 5964). 6.9. Pseudodifferential operators.
Pseudodifferential operators are generalizations of differential operators and singular integrals. They are formally defined as in equation (4.17), that IS,
where CJ, the symbol of T, is a complexvalued function defined on ~n X ~n. Conditions must be imposed on CJ to ensure that T f is well defined for
f
E cgo(~n).
If CJ(x, e) = m(e) is independent of x, then T is the multiplier associated with m; if CJ(X, = b(x) is independent of T corresponds to pointwise multiplication by b. Although in these cases T is bounded on L2 if and only if m or b are bounded functions, this is not true in general. For example, CJ(x, e) = b(x )m(e) with m, b E L2 can be unbounded, but by applying Holder's inequality it is immediate that T is always bounded on L2.
e)
e,
Symbols are classified according to their size and the size of their derivatives. The simplest classification is motivated by the study of elliptic differential operators: given m E Z, we say that CJ E if
sm
ID~DrCJ(x, e)1 < Ca ,jJ(1 + lel)m1a l . Pseudodifferential operators can be rewritten as
(5.24)
Tf(x)
=
r K(x, x  y)f(y) dy,
J~n
where K is given by (4.19). When CJ E SO then one can show that K satisfies a Hormander condition (cf. (5.10) and (5.11)) and that T is bounded on L2.
5. Singular Integrals (II)
114
Hence the theory developed in this chapter applies and T is bounded on LP, 1 < p < 00. The composition of pseudodifferential operators yields a corresponding symbolic calculus. In the simplest cases (pointwise multiplication and multipliers) the symbol of the composition of two operators is the product of the symbols. (Cf. Theorem 4.15.) In general, given pseudodifferential operators T1 and T2 with symbols in sml and 8 m2 , their composition is a pseudodifferential operator with symbol in sml +m2 . Further, the symbol of T1 0 T2 has an asymptotic expansion whose dominant term is the product of the symbols of T1 and T2 and the remaining terms are symbols of pseudodifferential operators in 8 m1 +m 2  j , j > O. If (J E 8 m is an elliptic operator (that is, it has a lower bound 1(J(x,~) I > C(l + 1~I)m), then there exists a symbol in such that the composition of the corresponding operators is the identity (which is in 8 0 ) plus an error term with symbol in 8 1 . Thus, the theory of pseudodifferential operators with symbols in is readily applicable to the problem of the inversion of elliptic operators.
sm
sm
When considering the corresponding problem for other differential operators, a more general class of symbols arises. We say that (J E s~o if ID~D~(J(x, ~)I
< A a,/3(l + 1~I)mplal+oIJ31.
This class includes the previous one: with this notation 8 m becomes Slo. , If T is a pseudodifferential operator with symbol in S~o' then it can again be written in the form (5.24). One can show that the kernel K satisfies the standard estimates (5.12), (5.13) and (5.14) if and only if m = 0 and p = 6 < 1. A remarkable result due to A. P. Calderon and R. Vaillancourt (A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 11851187) gives us that a pseudodifferential operator with symbol in 82,p for some 0 < P < 1 is bounded on L2, and so we can again apply the techniques of this chapter to show that it is bounded on LP. The proof of Calderon and Vaillancourt uses Cotlar's lemma (see Chapter 9, Section 1). Their result is false when p = 1: there exist symbols in 8~,1 (and so in L OO ) whose associated pseudodifferential operators are unbounded on L2.
When m =f 0, the properties of pseudodifferential operators whose symbols are in S~o are studied using a scale of spaces which measure the regularity of functions; for example, the Sobolev spaces (see Chapter 4, Section 7.7) or the Lipschitz spaces (see Stein [15, Chapter 5]). For further results on pseudodifferential operators and additional references see Stein [17], Journe [8] and CoHman and Meyer [2].
Chapter 6
HI and BMO
1. The space atomic HI In Chapter 3, Section 3, we saw that the Hilbert transform of an integrable function is not, in general, in Ll, and that a necessary condition for this is that the function have zero integral. Here we are going to define a subspace of Ll whose image under any singular integral is in Ll. We begin by defining its basic elementsatoms.
Definition 6.1. An atom is a complexvalued function a defined on which is supported on a cube Q and is such that
!aa(X)dx=O
}Rn
1
Iiall oo < 1Qf'
and
Proposition 6.2. Let T be an operator as in the hypotheses of Theorem 5.10 whose kernel satisfies condition (5.10). Then there exists a constant C such that, given any atom a,
IITalh < C.
Proof. Since a E L2, Ta is well defined. Let Q* be the cube with the same center as Q, cQ, and side length 2fo times larger. Then, since T is bounded on L2 , 1/2
!a. ITa(xll dx < IQ'1
1/ 2 (
!a. ITa(xW dx
)

115
6. HI and BMO
116
< ClQI'/2
(10 la(xll 2dX) '/2
~C.
Further, since a has zero average, by condition (5.10)
r
J~P\Q*
ITa(x)1 dx
r r K(x,y)a(y) dy dx = r r [K(x, y)  K(x, cQ)]a(y) dy J'Rn\Q* JQ =
J~n\Q* JQ
rr
<
JQ J'Rn\Q*
dx
IK(x, y)  K(x, cQ)1 dx la(y)1 dy
o We now define the space atomic HI, denoted by Hlt, by
H~t(lRnl
= {
~>jaj
: aj
atoms,
Aj E
C,
~ IAjl <
00 } •
Clearly, Hlt is contained in Ll. Define a norm on Hlt by
IlfIIH~,
= inf {
~ IAjl : f L =
Ajaj } .
\Vith this norm Hlt is a Banach space. (Details are left to the reader.) Further, it follows immediately from Proposition 6.2 that singular integrals are bounded from Hlt to Ll. Corollary 6.3. Let T be an operator as in Proposition 6.2, and let f E Hlt .
Then
IITflh < CllfllH~t· The space Hlt is the largest subspace of Ll(I~n) for which Corollary 6.3 holds in the following sense: let R l , ... ,Rn be the Riesz transforms in ~n (the Hilbert transform if n = 1), and define the space Hl(~n) =
{f
E Ll(~n)
: Rjf
E Ll(~n), 1 <j
with the norm n
IIfllHl =
IIfill
+L j=l
Then the following is true.
IIRjflll.
< n}
117
2. The space BMO
We will not prove Theorem 6.4; a proof can be found in either GarciaCuerva and Rubio de Francia [6, Chapter 3] or Stein [17, Chapter 3].
2. The space BMO Given a function f E Lfoc(lRn) and a cube Q, let fQ denote the average of f on Q:
fQ =
I~I far
Define the sharp maximal function by
(6.1)
M# f(x)
= sup IQ11 Q3x
i
r If  fQ!, Q
where the supremum is taken over all cubes Q containing x. Each of these integrals measures the mean oscillation of f on the cube Q; we say that f has bounded mean oscillation if the function M# f is bounded. The space of functions with this property is denoted by BMO:
BMO = {f E Lfoc : M# f E LOO}. We define a norm on BMO by
11111*
= IIM# flloo.
This is not properly a norm since any function which is constant almost everywhere has zero oscillation. However, these are the only functions with this property, and it is customary to think of BMO as the quotient of the above space by the space of constant functions. In other words, two functions which differ by a constant coincide as functions in BMO. On this space 11·11* is a norm and the space is a Banach space. It is easy to see that we have the pointwise inequality
M# f(x) < CnMf(x) with the HardyLittlewood maximal function. The constant depends on the dimension n, but if we replace M by M", the variant of the maximal operator where the supremum is taken over all cubes containing x (see (2.6)), the constant is 2. Proposition 6.5.
(6.2)
(6.3)
1 . 1 Ilfll* < sup mf IQI 2 Q aEC
1 Q
If(x) 
al dx < Ilfll*;
M#(lfl)(x) < 2M# f(x).
6. HI and BMO
118
Proof. The second inequality in (6.2) is immediate. To prove the first, note that for all a,
k
If(x)  fQldx:;
k
If(x)  al dx +
Now divide both sides by supremum over Q.
IQI,
k
la 
k
If(x)  al dx
take the infimum over all a and then the
Inequality (6.3) follows from (6.2) (with a
I~I
fol dx < 2
k
Ilf(x)II!QII dx <
I~I
=
IfQI) since
k
If(x)  fQI dx.
o
"*'
Inequality (6.2) defines a norm equivalent to " . one which provides a way to show that f E BMO without using its average on Q: it suffices to find a constant a (that can depend on Q) such that
I~I
k
If(x)  al dx < C
with C independent of Q.
It follows from (6.3) that if f E BMO then If I is also in BMO. However, the converse is not true. This confirms what the definition suggests: being in BMO is not just a question of size. Clearly Loo C BMO, but there are also unbounded BMO functions. The typical example on IR. is
f(x)
log ( 1) = {0 Ixl
Ixl < 1,
IX,I > 1. But it is easy to see that the function sgn(x)f(x) ¢ BMO even though f is its absolute value. The next result shows the connection between BMO and singular integrals. Theorem 6.6. Let T be an operator as in the hypotheses of Theorem 5.10 whose kernel satisfies condition (5.11). Then if f is a bounded function of compact support, Tf E BMO and
Proof. Fix a cube Q in IR. n with center cQ, and let Q* be the cube centered at cQ whose side length is· 2.Jii, times that of Q. Decompose f as f = h + 12, where h = fXQ*·
119
2. The space BMO
Let a
= T 12 (cQ). Then, since (5.11) holds and since
L2 ,
I~I
k
ITf(x) 
<
T is bounded on
al dx
kITh{x)ldx + I~I k C~I k f2 I~I
ITj,(x)  Tj,(cQ) I dx
<
I
T h(x)1 2 dx
+ IQ11 ( {
JQ J'Rn\Q*
C~I
k.
[K(x, y)  K(cQ, y)]j(y) dy dx
If(x)12 dx )
+ IQ11 ( {
JQ J'Rn\Q*
1/2
IK(x, y)  K(cQ' y)1 dy dx· 11J1100
< CllJlloo. D Theorem 6.6 is a first step towards our goal of finding a space which contains the images of bounded functions under singular integrals. However, the set of bounded functions with compact support is not dense in L oo , so we cannot extend T to the whole space by continuity. Instead, we will extend the definition of T to one which holds on all of Loo. Let f be a bounded function and let Q be a cube in ]Rn centered at the origin. Define Q* as above and again decompose f as h + 12, h = fXQ*· Since h is bounded and has compact support, This well defined as an L2 function; hence T h (x) exists for almost every x. For x E Q define
(6.4)
Tf(x)
=
Th(x)
+ (
J'Rn
[K(x, y)  K(O, Y)]12(Y) dy.
This integral converges since it is bounded by
Ilflloo (
J'Rn\Q*
IK(x, y)  K(O, y)1 dy,
provided K satisfies (5.11). N ow let Q be sOII?e other cube centered at the origin which contains Q. Then we have two definitions of Tf(x) for x E Q. Let 11 = fX{r and 12 = f X'Rn \{j*; then the difference between the two definitions is
6. HI and BMO
120
r [K(x, y)  K(O, y)Jf(y) dy  r _[K(x, y)  K(O, y)Jf(y) dy =  (
T(h  JI)(X)
+
JJRn\Q*
JJRn \Q*
K(O, y)f(y) dy,
J Q*\Q*
and this is independent of x. Hence, the two definitions coincide as functions in BMO since in BMO we identify functions which differ by a constant. We can, therefore, take (6.4) as the definition of the image of f under T. Further, by arguing as we did in Theorem 6.6, we can show that if f E L oo then T f E BMO. Finally, note that we do not have to take cubes centered at the origin but can take any cube. If f is a bounded function of compact support, then (6.4) coincides with our original definition since for Q large enough, f = h. More generally, if f E S then the two definitions agree as B M 0 functions: since
r
K(O, y)f(y) dy
JJRn\Q*
converges absolutely to a value independent of x, the two definitions differ by a constant. Example 6.7. Let f(x) = sgn(x); we will find the image of f under the Hilbert transform, whose kernel is K(x, y) = [7f(x  y)]I. Let Ixl < a/2; then
. ja sgn(y) dy +
7fHf(x) = p. v.
a X 
Y
lim
ja + iN (1   + 1)
N+oo
N
a
X 
Y
Y
sgn(y) dy
= 2log Ixl  2Iog(a). Hence, ignoring the constant,
H(sgn(x))
=
2
log Ixl. 7f
This is an indirect proof that log Ixl is a function in BMO. This result also agrees with our original motivation for the Hilbert transform since there exists an analytic function in the upper halfplane whose real part tends to sgn(x) and whose imaginary part tends to ~ log Ixl as we approach the real axis. We can .give such a function explicitly: i log(iz) As y
+
= i log( y + ix) = i log(x 2 + y2)1/2 + arctan(x/y).
0 the limit of this function is 7f
.
 sgn(x) + dog Ixl. 2 .
3. An interpolation result
121
3. An interpolation result In applying interpolation theorems (such as the Marcinkiewicz interpolation theorem) we often use the fact that an operator is bounded from L= to L=. Here we show that we can replace this with the weaker condition of boundedness from L= to BMO. Theorem 6.8. Let T be a linear operator which is bounded on LPO for some Po, 1 < Po < 00, and is bounded from L= to BMO. Then for all p, Po < p < 00, T is bounded on LP.
The proof of Theorem 6.8 requires two lemmas. Lemma 6.9. If 1
(6.5)
00
and f E LPO then
r Mdf(x)P dx < c JlRnl M# f(x)P dx,
JlRn
where Md is the dyadic maximal operator (2.9). By Lemma 2.12 we can replace the dyadic maximal function in (6.5) with the HardyLittlewood maximal function. Thus, while the reverse of the pointwise inequality M# f(x) < CM f(x) is not true in general, Lemma 6.9 provides a substitute norm inequality. The proof of Lemma 6.9 uses a technique known as a "goodA inequality"; more precisely, it depends on the following result. Lemma 6.10. If f E LPO for some Po, 1
<
00,
then for all "f > 0 and
A> 0, I{x E JRn : Mdf(x) > 2A, M# f(x) < "fA} I < 2n yl{x E JRn : Mdf(x) > A}I.
Proof. Without loss of generality we may assume that f is nonnegative. Fix A, y > o. If we form the Calder6nZygmund decomposition of f at height A, then the set {x E JRn : Mdf (x) > A} can be written as the union of disjoint, maximal dyadic cubes. (See Theorem 2.11; we can substitute f E LPO for the hypothesis fELl in the proof.) If Q is one of these cubes, then it will suffice to show that
Let Q be the dyadic cube containing Q whose sides are twice as long. Then, since Q is maximal, fa < A. Furthermore, if x E Q and Mdf(x) > 2A, then Md(fXQ)(x) > 2A, so for those x's,
Md((f  fo)XQ)(x) > Md(fXQ)(x)  fa > A.
6. HI and BMO
122 By the weak (1,1) inequality for Md (Theorem 2.10),
I{x E Q : Md((f  J(j)xQ)(x) > A}I <
~ 10 IJ(x)  J¢I dx
2n lQI 1 ( < AIQI lQ If(x) 
fol dx
2n lQI < inf M# f(x).
A
xEQ
If the set {x E Q : Mdf(x) > 2A, M# f(x) < ,,(A} is empty there is nothing to prove. Otherwise, for some x E Q, M# f(x) < "(A and so inequality (6.6) holds. 0 Proof of Lemma 6.9. For N
> 0 let
IN = ioN pAP'I{x E]Rn : MdJ(X) > A}I dA. IN is finite since IN < and Mdf E
~NPPO
(N POAPOIl{x E lRn Po lo £PO since f E £Po. Furthermore,
:
Mdf(x) > A}I d,X,
IN = 2 io N p>'P'I{x E]Rn : MdJ(X) > 2A}1 dA P
!2
{N/2
< 2Plo'
p,Xp1 (I{x E lR n
:
Mdf(x) > 2A, M# f(x) < "(A}I
+ I{x E lRn 2P 1'YN/2
< 2P+n ,,(IN + 
(6.7)
P I IN < 2 + "(P
M# f(x) > "(A}I)dA
, p,Xpll{x E lR n : M# f(x) > A}I d'x.
0
"(P
Fix"( such that 2P+n ,,(
:
= 1/2; then
l'Y
N/ 2
p,Xpll{x
E
lR n
:
0
M# f(x) > A}I d'x.
If f (M# f)P = 00 then there is nothing to prove. If it is finite then we can take the limit as N ~ 00 in inequality (6.7) to get (6.5). 0 Proof of Theorem 6.8. The composition M# 0 T is a sublinear operator. It is bounded on LPO since both these operators are, and it is bounded on L oo since
IIM#(Tf)lJoo = IITfll* <
Cllflloo.
Hence, by the Marcinkiewicz interpolation theorem M# LP, Po < P < 00.
0
T is bounded on
4. The JohnNirenberg inequality
123
Now suppose that f E LP has compact support. Then f E LPo and so Tf E LPo. Therefore, we can apply Lemma 6.9 to Tf. In particular, since ITf(x)1 < Md(Tf)(x) a.e.,
r
J~n
ITf(x)I P dx <
r
J~n
Md(Tf) (x)P dx
r M#(Tf)(x)P dx < 0 r If(x)IP dx. J~n <0
J~n
o 4. The JohnNirenberg inequality In this section we examine the rate of growth of functions in B MO. We first consider 10g(I/lxl), which we gave as an example of an unbounded function in BMO. On the interval (a, a), its average is (Ilog a); given A > 1, the set where Ilog(I/lxl)  (1  log a) I > A has measure 2ae A 1 . The next result shows that in some sense logarithmic growth is the maximum possible for BMO functions. Theorem 6.11 (JohnNirenberg Inequality). Let f E BMO. Then there exist constants 0 1 and O2 , depending only on the dimension, such that given any cube Q in IR n and any A > 0,
(6.8)
I{x E Q : If(x)  fQI > A}I < 0Iec2A/llfll* IQI·
Proof. Since (6.8) is homogeneous, we may assume without loss of generality that IIfll* = 1. Then
(6.9)
I~I
k
If(x)  fQI dx < 1.
Now form the Calder6nZygmund decomposition of f  fQ on Q at height 2. (This is done as in Theorem 2.11 except that we begin by bisecting the sides of Q to form 2n equal cubes and repeating this for all cubes formed. Inequality (6.9) replaces the hypothesis that f ELI.) This gives us a family of cubes {Ql,j} such that 2 < IQl .1 I,J
r
JQl,j
If(x)  fQI dx < 2n+l
6. HI and BMO
124
and If(x)  fQI < 2 if x
rJ Uj Q1,j.
In particular,
:L. IQ1,jl < ~ JrQ If(x) 
fQI dx <
~IQI,·
J
and
IfQu  fQI
= IQ1
.1
1,J
r
JQ1
(f(x)  fQ) dx < 2n+1. ,j
On each cube Q1,j form the Calder6nZygmund decomposition of f fQl,j at height 2. (Again by assumption the average of this function on Q1,j is at most 1.) We then obtain a family of cubes {Q1,j,k} which satisfy the following:
IfQl,j,k  fQl,j I < 2n+1,
UQ1,j,k,
If(x)  fQl,j I <2 if x E Q1,j \
k
:L IQ1,j,kl < "21 IQ1,jl. k
Gather the cubes Q1,j,k corresponding to all the Q1,j'S and collectively rename them {Q2,j}. Then
and if x
rJ Uj .Q2,j,
If(x)  fQI < If(x)  fQl,jl
+ IfQl,j
 fQI < 2 + 2n+1 < 2· 2n+1.
Repeat this process indefinitely. Then for e<;Lch N we get a family of disjoint cubes {Q N,j} such that
If(x)  fQI < N· 2n+1 if x
rJ Uj QN,j
and
Fix A > 2n+1 and let N be such that N2 n+1 < A < (N
I{x
E
Q : If(x)  fQI > A}I <
:L IQN,jl j
1
< 2NIQI = eNlog2lQI < e C2A IQI,
+ 1)2n+1.
Then
4. The JohnNirenberg inequality
125
where C2 = log2/2n+2. If A < 2n+1 then C 2 A < 10gv'2. Hence,
I{x
E
Q : If(x)  fQI > A}I < IQI < elogv!2C2AIQI = V2e C2A IQI,
so we can take C 1
o
= v'2.
Corollary 6.12. For all p, 1 < p <
Ilfll.,p =
00,
s~p (I~I 10 If(x) II . Ik
is a norm on BMO equivalent to
Proof. It will suffice to prove that Ilfll*,p < equality is immediate. But by Theorem 6.11
10 If(x)  fQI Pdx
=
1''
I~I 10 If(x) 
Cpllfll*
since the reverse in
p).pll{x E Q : If(x)  fQI > )'}I d)'
< C11QI
Make the change of variables s
fQI PdX) 'ip
10
00
pAP1eC2A/llfll* d)".
r10=
= C2A/llfll*; then we get
fQIP dx < C,p
e~~'
r'e s ds
= c1PC;Pr(p)lIfll;,
o
which yields the desired inequality.
As a consequence of the proof of Corollary 6.12 we get two additional results. Given the size of the constant Cp , if we expand the exponential function as a power series we immediately get the following. Corollary 6.13. Given f E BMO, there exists).. cube Q,
>
0 such that for any
Further, the proof of Corollary 6.12 (with p = 1) can be readily adapted to show that the converse of the JohnNirenberg inequality (Theorem 6.11) holds. Corollary 6.14. Given a function f, suppose there exist constants C 1 , C2 and K such that for any cube Q and A > 0,
I{x Then f E BMO.
E
Q : If(x)  fQI > )..}I < C1eC2A/KIQI·
6. HI and BMO
126
5. Notes and further results 5.1. References.
For the development of Hardy spaces and in particular the space HI, see Section 5.2 below. E. M. Stein (Classes HP, multiplicateurs et fonctions de LittlewoodPaley, C. R. Acad. Sci. Paris 263 (1966), 716719, 780781) first proved that singular integrals and multipliers are bounded from HI to LI. The space BMO was introduced by F. John and L. Nirenberg while studying PDE's (On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415426). There they proved Theorem 6.11. That singular integrals take L= into BMO was first observed by S. Spanne (Sur l'interpolation . entre les espaces £~' ,Ann. Scuola Norm. Sup. Pisa 20 (1966), 625648), J. Peetre (On convolution operators leaving £p,).. spaces invariant, Ann. Mat. Pura Appl. 72 (1966), 295304), and E. M. Stein (Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 316335, Amer. Math. Soc., Providence, 1967). The sharp maximal function was introduced by C. Fefferman and E. M. Stein (HP spaces of several variables, Acta Math. 129 (1972), 137193); in the same paper they proved Theorem 6.8. 5.2. The HP spaces.
The Hardy spaces were first studied as part of complex analysis by G. H. Hardy (The mean value of the modulus of an analytic function, Proc. London Math. Soc. 14 (1914), 269277). An analytic function F in the unit disk JD) = {z E C : Iz I < I} is in HP (JD)), 0 < p < 00, if sup O
j7r IF(reiB)IP de < 7r
00.
A function F which is analytic in the upper halfplane y > O} is in HP(ffi.t) if sup y>O
r IF(x + iy)IP dx <
IIR
ffi.t = {z = x + iy :
00.
When p > 1 it can be shown that HP coincides with the class of analytic functions whose real parts are the Poisson integrals of functions f in £P(1I') or LP(ffi.) , respectively. We can therefore identify HP(lOl) with LP(1I') and HP(ffi.t) with LP(ffi.). This identification does not hold, however, for p < 1. Functions in HP(lOl) and HP(ffi.t) have a rich structural theory based on "factorization" theorems which allow them to be decomposed into well understood components. For a thorough treatment from this perspective, see Koosis [11] or Rudin [14]. Unfortunately, these results cannot be extended to higher dimensions using the theory of functions of several complex variables. E. M. Stein
5. Notes and further results
127
and G. Weiss (On the theory of harmonic functions of several variables, 1: The theory of HP spaces, Acta Math. 103 (1960), 2562, also see Stein [15, Chapter 7]) defined HP(JR~+I) by considering vectorvalued functions which satisfy generalized CauchyRiemann equations. This extension is only valId if p > 1  lin, which does include HI. This space HI is isomorphic (by passing to boundary values) to the space of functions in LI(JRn ) whose Riesztransforms are also in LI(JR n ). (This is the space we called HI (JR. n ) in Section 1.) D. Burkholder, R. Gundy and M. Silverstein (A maximal characterization of the class HP, Trans. Amer. Math. Soc. 157 (1971), 137153) proved a crucial result for establishing a strictly realvariable theory of HP spaces. Given F = u + iv, analytic in the upper halfplane, they showed that F is in HP if and only if the nontangential maximal function of its real part, u~(XO)
= sup{lu(x, y) I : (x, y)
E
r a(XO)},
where
r a(XO) = {(x, y)
E JR~ : Ix  xol < ay},
is in LP(JR.). (See Chapter 2, Section 8.7.) Since u(x, y) = P y * f(x), the Poisson integral of its boundary value f, this characterizes the restriction to JR. of the real parts of functions in HP (JR~) . This space of functions on JR., which is customarily denoted by HP(JR.) (or sometimes by Re HP(JR)), is equivalent when p = 1 to the space HI (JR.) defined in Section 1. The original proof of this result used Brownian motion; a complex analytic proof was given by P. Koosis (Sommabilite de la fonction maximale et appartenance a HI, C. R. Acad. Sci. Paris 28 (1978), 10411043, also see [11, Chapter 8]). The key feature of this result is that the fact that u is a harmonic function is irrelevant, in the sense that the same characterization of the boundary values holds if we replace the Poisson kernel by any approximation of the identity defined starting from ¢ E S with nonzero integral. This leads to an elegant definition of HP in higher dimensions: given ¢ E S(JR. n ) with nonzero integral, we say a function f E HP(JR. n ) if Mqd(xo)
= sup{l¢y * f(x) I : (x, y)
E
r a(XO)}
is in LP(JR.n ). The resulting space is independent of ¢: in fact, one can replace M¢ by the socalled grand maximal function Mf(x)
= supM¢f(x), ¢
where the supremum is taken over all such ¢. For p > 1  1/n this space is isomorphic (again by taking boundary values) to the HP spaces of Stein and Weiss above. This approach is developed in the excellent article by Fefferman and Stein cited above. Also see Stein [17, Chapter 3].
6. HI and BMO
128
The atomic decomposition of functions in HP(lR) is due to R. Coifman (A real variable characterization of HP, Studia Math. 51 (1974),269274); it was extended to higher dimensions by R. Latter (A decomposition of HP(n~n) in terms of atoms, Studia Math. 62 (1978), 93101; also see R. Latter and A. Uchiyama, The atomic decomposition for parabolic HP spaces, Trans. Amer. Math. Soc. 253 (1979),391398). For simplicity we used an atomic decomposition as the basis of our definition of HI (lRn )the space we called Hlt(lR n ). Using these methods, R. Coifman and G. Weiss (Extensions of Hardy spaces and their use in Analysis, Bull. Amer. Math. Soc. 83 (1977), 569645) defined HP spaces on spaces of homogeneous type (see Chapter 5, Section 6.4). Their definition only holds for Po < p < 1, where Po is a constant which depends on the space being considered. A monograph on the real theory of HP spaces with applications to Fourier analysis and approximation theory is due to Shanzhen Lu (Four Lectures on Real HP Spaces, World Scientific, Singapore, 1995). Finally, we mention another characterization of HP spaces in terms of the Lusin area integral. If u is a harmonic function on lR~+I, let
where
~~(xo)
=
{(x, y) E lR~+1 :
Ix 
xol
< ay, 0 < y <
h}
and
(This is called the area integral since if n = 1, S (u ) 2 is the area of the image of r~(xo), counting multiplicities, under the analytic function F whose real part is u.) Then u E HP(lR~+I) if and only if S(u) E LP(lRn ) and u(x,y) t 0 as y t 00. An alternative characterization is gotten by replacing the area integral by a LittlewoodPaley type function which is its radial analogue: g( u) (x)
=
(Joroo l\7u(x, y) 12y dy.) 1/2
This characterization is due to A. P. Calderon (Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 10921099). See also GarciaCuerva and Rubio de Francia [6], Stein t15], Zygmund [21] and the article by Fefferman and Stein cited above.
5. Notes and further results
129
5.3. Duality.
The results on boundedness from HI to LI and from L= to BMO are dual to one another. This is a consequence of a deep result due to C. Fefferman (announced in Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587588). Theorem 6.15. The dual of HI (JR n ) is BMO.
The article by Fefferman and Stein cited above gives two proofs of this result. Three appear in GarciaCuerva and Rubio de Francia [6]. The duals of the HP spaces, P < 1, are Lipschitz spaces. This is due to P. Duren, B. Romberg and A. Shields (Linear functionals on HP spaces with o < P < 1, J. Reine Angew. Math. 238 (1969), 3260) on the unit circle and to T. Walsh (The dual of HP(JR+.+ I ) for P < 1, Can. J. Math. 25 (1973), 567577) in JRn . Also see [6, Chapter 3]. 5.4. The HardyLittlewood maximal function on BMO.
The maximal function is bounded on BMO: if f E BMO then Mf E BMO. This was first proved by C. Bennett, R. A. DeVore and R. Sharpley (Weak L= and BMO, Ann. of Math. 113 (1981), 601611); simpler proofs were given by F. Chiarenza and M. Frasca (Morrey spaces and HardyLittlewood maximal function, Rend. Mat. Appl., Series 7, 7 (1981), 273279) and D. CruzUribe, and C. J. Neugebauer (The structure of the reverse Holder classes, Trans. Amer. Math. Soc. 347 (1995), 29412960). See also a proof in Torchinsky [19, p. 204]. 5.5. Another interpolation result. 1
In applying the Marcinkiewicz interpolation theorem one often uses that the operator in question is bounded on L= or satisfies a weak (1,1) inequality. In Theorem 6.8 we showed that we can replace the first hypothesis by the assumption that the operator maps L= into BMO. One can also use HI to replace the weak (1,1) inequality. Theorem 6.16. Given a sublinear operator T, suppose that T maps HI into LI and for some PI, 1 < PI < 00, T is weak (PbPI). Then for all p, 1 < P < PI, T is bounded on LP.
This result can be generalized to the other HP spaces. For details and references, see GarciaCuerva and Rubio de Francia [6, Chapter 3]. 5.6. Fractional integrals and Sobolev spaces.
For fractional integral operators fa (see Chapter 4, Section 7.7), the critical index is P = n/ a: we would expect fa to map L n / a to L=, but it does not. Instead, we have a result analogous to Theorem 6.6.
6. HI and BMO
130
Theorem 6.17. If f E Ln/a then Iaf E BMO.
The proof of this theorem follows at once from an inequality relating the fractional integral, the sharp maximal function and the fractional maximal function. Theorem 6.18. There exist positive constants C I and C 2 such that for all locally integrable f,
CIMaf(x)
< M#(laf)(x) < C 2 M a f(x).
Both of these results are due to D. R. Adams (A note on Riesz potentials, Duke Math. J. 42 (1975), 765778). When p = 1, besides the weak (1, n/(n  a)) inequality in Theorem 4.18, we also have an analogue of Corollary 6.3: IIlaflln/(na) < CllfllHl. This result is due to Stein and Weisssee the article cited above. Similar inequalities hold for all the HP spaces: see the article by S. Krantz (Fractional integration on Hardy Spaces, Studia Math. 73 (1982), 8794). For Sobolev spaces, the corresponding limiting theorem would be that if f E L~/k then f E BMO. However, this is only known when k = 1: see the paper by A. Cianchi and L. Pick (Sobolev embeddings into BMO, VMO and L oo , Ark. Mat. 36 (1998), 317340). In the general case we have as a substitute an exponential integrability result (cf. Corollary 6.13). Theorem 6.19. Given 0 < a < n and any cube Q, then there exist constants C I and C2 such that if f E Ln/a(Q),
1 IQI
_1
exp (
Q
laf ' PI) < CIllfIILn/a(Q) 
c2 .
Theorem 6.19 was proved by N. S. Trudinger (On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473483) when a = 1 and in general by R. S. Strichartz (A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841842). Sharp constants have been found by J. Moser (A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 10771092) and D. R. Adams (A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385398). Also see the books by W. Ziemer (Weakly Differentiable Functions, SpringerVerlag, New York, 1989) and D. R. Adams and L. Hedberg (Function Spaces and Potential Theory, SpringerVerlag, Berlin, 1996).
5. Notes and further results
131
5.7. Vanishing mean oscillation.
. Given a function f E BMO, we say that f has vanishing mean oscillation, and write f E V MO, if
I~I
fa
If(x)  fQI dx
tends to zero as IQI tends to either zero or infinity. As in BMO, there are unbounded functions in V MO. For example,
f(x)
=
{~~g log(l/Ixl), Ixl < lie, Ixl > lie,
is in VMO, and in fact it can be shown that in some sense it has the maximum rate of growth possible for functions in V MO. Clearly, if f E Co (i.e. the space of continuous functions which vanish at infinity) then f E V MO; further, V MO is the closure in BMO of Co.
V MO lets us sharpen Theorem 6.6 as follows: given a singular integral operator T, if f E Co then Tf E VMO. The space V MO was introduced by D. Sarason (Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391405). Also see the survey article by R. Coifman and G. Weiss (Extensions of Hardy spaces and their uses in analysis, Bull. Amer. Math. Soc. 83 (1977), 569645). 5.8. Commutators and BMO.
We can give another characterization of BMO in terms of singular integrals. Let T be a singular integral, and for a locally integrable function b, let Mb be the operator given by pointwise multiplication by b: Mbf(x) = b(x)f(x). Define the commutator [b, T] by MbT  TMb. If Tf = K * f then for f E C~ [b, T]f(x)
=
r
JlR
(b(x)  b(y))K(x  y)f(y) dy,
x Fj. supp(f).
n
Clearly, if b E L= then [b, T] is bounded on LP, 1 < p < 00. Since both MbT and T Mb are bounded exactly when b is a bounded function, it would be reasonable to conjecture that this is also the case for [b, T]. However, due to cancellation between the two terms, a weaker condition is sufficient. Theorem 6.20. Given a singular integral T whose associated kernel is standard, then the commutator [b, T] is bounded on LP, 1 < p < 00, if and only ifb E BMO.
Commutators were introduced by R. Coifman, R. Rochberg and G. Weiss (Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611635) who used them to extend the classical theory of HP
132
6. HI and BMO
spaces to higher dimensions. They proved that b E B M 0 is sufficient for [b, T] to be bounded and proved a partial converse. The full converse is. due to S. Janson (Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), 263270). This paper also contains a simpler proof of sufficiency due to J. O. Stromberg which is reproduced in Torchinsky [19, p.418]. The commutator [b, T] is more singular than the associated singular integral; in particular, it does not satisfy the corresponding weak (1,1) estimate. However, the following weaker result is true. Theorem 6.21. Given a singular integral Tand bE BMO, then
J{x E]Rn : J[b, TIf("')J > A}J < Cllbll*
Ln Jf~)J (1 + log+ (Jf~)J)) dy.
Theorem 6.21 is due to C. Perez (Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), 163185). In this paper he also discusses the boundedness of [b, T] on HI. A. Uchiyama (On the compactness of operators of Hankel type, T6hoku Math. J. 30 (1978), 163171) proved that the commutator [b, T] is a compact operator if and only if b E V MO.
Chapter 'l
Weighted Inequalities
1. The Ap condition In this chapter we are going to find the nonnegative, locally integrable functions w such that the operators we have been studying are bounded on the spaces V(w). (These are the LP spaces with Lebesgue measure replaced by the measure w dx.) We will refer to such functions as weights, and for a measurable set E we define w(E) = w.
IE
Our first step is to assume that the HardyLittlewood maximal function satisfies a weighted, weaktype inequality, and deduce from this a necessary condition on the weight w. To simplify our analysis, throughout this chapter we will replace our earlier definition of the HardyLittlewood maximal function with Mf(x) =
~~~ I~I 10 Ii(y) Idy,
where the supremum is taken over all cubes Q containing x. This is the noncentered maximal operator we originally denoted by Mil (see (2.6)), but recall that it is pointwise equivalent to our original definition (2.3). The weighted, weak (p, p) inequality for M with respect to w is (7.1)
W({XEIRn:Mf(x»A})<
~
AP
r If(x)IP
J'ffln
w (x)dx.
f
be a nonnegative fun~tion and let Q be a cube such that f(Q) = f > O. Fix A such that 0 < A < f(Q)/IQI. Then Q C {x E IR n : M(JXQ)(x) > A}, so from (7.1) we get that
Let
IQ
w(Q)
<
~ 10 If(xWw(x) dx.

133
134
7. Weighted Inequalities
Since this holds for all such A, it follows that (7.2) Given a measurable set SeQ, let
(7.3)
w(Q)
f = xs. Then (7.2) becomes
G~DP < eweS).
From (7.3) we can immediately deduce the following: (1) The weight w is either identically 0 or w > 0 a.e. For if w = 0 on a set of positive measure S (which we may assume is bounded), then (7.3) implies that for every cube Q that contains S, w(Q) = O.
(2) The weight w is either locally integrable or w = 00 a.e. For if w (Q) = 00 for some cube Q, then the same is true for any larger cube, and so by (7.3), w(S) = 00 for any set S of positive measure. Note that while we had assumed a priori that w was locally integrable, this is actually a consequence of the weighted norm inequality. To deduce the desired necessary conditions, we will consider two cases. Case 1: p
=
1. In this case inequality (7.3) becomes
w(Q) < Cw(S)
IQI 
lSI·
Let a = inf{w(x) : x E Q}, where inf is the essential infimum, that is, excluding sets of measure zero. Then for each E > 0 there exists SE C Q such that ISEI > 0 and w(x) < a + Efor any x ESE. Hence, for all E> 0,
w(Q)
lQf < C(a + E), and so
w(Q)
.
lQf < C ~~b w(x). Therefore, for any cube Q,
(7.4)
w(Q) . lQf < Cw(x)
a.e. x E Q.
This is called the Al condition, and we refer to the weights which satisfy it as Al weights. Condition (7.4) is equivalent to '
(7.5)
Mw(x) < Cw(x)
a.e. x E
]Rn.
Clearly, (7.5) implies (7.4). Conversely, suppose that (7.4) holds and let x be such that Mw(x) > Cw(x). Then there exists a cube Q with rational vertices such that w(Q)/IQI > Cw(x), so x lies in a subset of Q of measure
1. The Ap condition
135
zero. Taking the union over all such cubes we see that Mw(x) > Cw(x) holds only on a set of measure 0 in IRn.
Case 2: 1
< p < 00. In (7.2) let f = wIp' XQ; then w(Q)
(_1 {wIPI)P < C { wIpI. IQI J J Q
Q
Equivalently, (7.6) where C is independent of Q. (Note that to derive (7.6) we have assumed that wIp' is locally integrable. To avoid this assumption we can replace it by min (wIpI ,n) and take the limit as n tends to infinity. However, since w > 0 a.e., (7.6) implies that wIpI is locally integrable.) Condition (7.6) is called the Ap condition, and the weights which satisfy it are called Ap weights. Theorem 7.1. For 1
00,
the weak (p,p) inequality
w({xEIRn:Mf(x»A}) <
~ (
AP
JlR
If(x)IPw(x)dx
n
holds if and only if w E Ap. Proo[ We proved the necessity of the Ap condition above. To prove sufficiency, we first consider the case p = 1. This case is a corollary to Theorem 2.16. The righthand side of the weak (1,1) inequality (2.11) contains Mw, and since wE AI, by (7.5) we can replace Mw by Cw. Now suppose that p > 1 and w E Ap. Given a function f, we first show that inequality (7.2) holds, and so inequality (7.3) also holds. By Holder's inequality
Now fix f E LP(w); we may assume without loss of generality that f is nonnegative. We may also assume that fELl since otherwise we can replace f by fXB(O,k) and the following argument will yield constants independent of k. (Note that the previous argument shows that f is locally integrable.) Form the Calder6nZygmund decomposition of f at height 4 n A to get a collection
136
7. Weighted Inequalities
of disjoint cubes {Qj} such that f(Qj) > 4 n.\IQjl. (See Theorem 2.11.) Then by the same argument as in the proof of Lemma 2.12, we can show that
c U3Qj.
{x E R n : Mf(x) > .\}
j
(Here we dilate the cubes by a factor of 3 instead of 2 because M is the noncentered maximal operator.) Therefore,
w({x
E]Rn:
Mf(x) > .\}) < Lw(3Qj) j
j
< C3 np L ( IQjl )P f(Qj)
j
< C3 np (
~
rL
r IfIPw
JQj
IflPw,
where the second inequality follows from (7.3), the third from (7.2) and the fourth from the properties of the Q/s. 0 The following properties of Ap weights are consequences of the definition. Proposition7.2.
(1) Ap c Aq, 1 < p < q. (2) w E Ap if and only if wIpI E ApI. (3) If WO, WI E Al then wowi p E Ap. Proof. (1) If p
If ( IQI J
= 1 then
_ ,)qI
Q wI q
< ~~g w(x)
_
(.
1
= ~~£ w(x)
)1 < C (W(Q))I lQI
If p > 1 then this follows immediately from Holder's inequality. (2) The ApI condition for WI_pI is (
_1_
r wIpI)
IQI JQ
and since (p'  l)(p  1) to the power p'  1.
(
_1_
r W(1PI)(IP))
IQI JQ
pi 1

,
= 1, the lefthand side is the Ap condition raised
2. Strongtype inequalities with weights
137
(3) We need to prove that (
(7.7)
1
JQJ
fo
Ip )
wow,
(
1
JQJ
fo
PlI
I~
Wo
)
'S C.
w,
By the Al condition, for x E Q and i = 0,1,
Wi(X)I < SUpWi(X)I = (inf Wi(X))I < C xEQ
xEQ
(WI(~))I Q
If we substitute this into the lefthand side of (7.7) for the negative exponents we get the desired inequality. 0
2. Strongtype inequalities with weights Using the Marcinkiewicz interpolation theorem, we can immediately derive a strong (p, p) inequality from Theorem 7.1. Let W E A q , 1 < q < p. Then Loo(w) = L oo with equality of norms since by (7.3), w(E) = 0 if and only if lEI = O. Therefore, we have the inequality IIMfllLoo(w) < IIfllLOO(w)· If we interpolate between this and the weak (q, q) inequality we get
J.
(7.8)
Rn
1M flPw
< Cp
r IflPw
iRn
for all p > q. A much deeper result is that this inequality remains true when p = q. Theorem 7.3. If 1 < p < W
ex)
then M is bounded on LP(w) if and only if
EAp.
Since the strong (p, p) inequality implies the weak(p, p) inequality, necessity follows from Theorem 7.1. Sufficiency would follow from the above interpolation argument if we could show that given W E A p , there exists q, 1 < q < p, such that W E A q . The existence of such a q comes from the following key result. Theorem 7.4 (Reverse Holder Inequality). Let W E A p , 1 < p < ex). Then there exist constants C and E > 0, depending only on p and the Ap constant of w, such that for any cube Q, (7.9)
(
_1
r
IQI i Q W
I+E) I/(I+E) < ~ 
r .
IQI i Q w.
The name of Theorem 7.4 comes from the fact that the reverse of inequality (7.9) is an immediate consequence of Holder's inequality. To prove this result we first need to prove a lemma.
138
7. Weighted Inequalities
Lemma 7.5. Let w E A p , 1
(1  :~:r <
C(w(Q)  w(S)).
181 < alQI then w(8) < C 
(~ a)P w(Q),
which gives us the desired result with
/3 =
1  C 1 (1  a)p.
o
Proof of Theorem 7.4. Fix a cube Q and form Calder6nZygmund decompositions of w with respect to Q at heights given by the increasing sequence w(Q)/IQI = AD < Al < ... <,Ak < ... ; we will fix the Ak'S below. (The decomposition is formed by repeatedly bisecting the sides of Q; cf the proof of Theorem 6.11.) For each Ak we get a family of disjoint cubes {Qk,j} such that
w(x) < Ak if x r:f. Ak < IQl
j
.11
k,J
.ok = UQk,j, w < 2nAk·
Qk,j
It is clear from their construction that nk + 1 C .ok. If we fix Q k,jo from the Calder6nZygmund decomposition at height Ak, then Qk,jo n nk+1 is the union of cubes Qk+l,i from the decomposition at height Ak+l. Therefore,
IQk,jo n nk +1 1 =
L
IQk+l,il '
i
<_1
r
w
 Ak+l JCQk ,10. 2nAk < ,IQk,jol· Ak+l Fix a < 1 and choose the Ak'S so that 2n Ak I Ak+ 1 (2 n a 1 )kw(Q)/IQI. Then IQk,jo n nk+11 < aIQk,jol,
/3 < 1 such that W(Qk,jo n nk+l) < /3W(Qk,jo)·
so by Lemma 7.5 there exists
a; that is, Ak
139
2. Strongtype inequalities with weights
Now sum over all the cubes in the decomposition at height Ak; we then get W(f2 k +l ) < f3W(f2k). If we iterate this inequality we get W(f2k) < f3k w (n o). Similarly, Inkl < aklnol; hence,
Therefore, 1
IQI
1
w l +E
= 1
IQI
Q
l
w l +E
Q\Oo
+ 1 L IQI k=O 00
1
w l +E
Ok\Ok+l
\Ew(Q) 1 ~\E (1"1) < "'olQl + IQT ~ "'k+I w Hk k=O
= AE w(Q) + ~ ~(2naI)(k+I)EAE 4kW(f2
0IQI
IQI ~
OfJ
k=O
)
0.
Fix E > 0 such that (2 n a 1 )Ef3 < 1; then the series converges and the last term is bounded by CA6w(Q)/IQI. Since AO = w(Q)/IQI we have shown the desired inequality. D As a corollary to the reverse Holder inequality we get the property of Ap weights needed to complete the proof of Theorem 7.3. Corollary 7.6.
= Uq
0 such that given a cube
(1) Ap
E
Q and SeQ, w(S)
(7.10)
< C (.~)8
w(Q) 
IQI
If a weight w satisfies (7.10) then we say that w E Aoo. We use this notation since then (1) holds with p = 00. See Section 5.3 below.
Proof. (1) If w E Ap then by Proposition 7.2, wIpI E ApI. Therefore, it also satisfies the reverse Holder inequality for some E > 0:
(7.11)
C~I
k
')(1+,») 1/(1+,) < I~I
w(lP
k
w '  p '.
Fix q such that (q'  1) = (p'  1)(1 + E); then q < p and inequality (7.11) together with the Ap condition implies that w E A q.
7. Weighted Inequalities
140
(2) If p > 1 then this follows immediately if we choose E > 0 small enough so that both wand wIpi satisfy the reverse Holder inequality with exponent 1 + E. If p = 1 then for any cube Q and almost every x E Q,
1
IQI
1
(IQI 1
wl +E < C
Q
w )I+E
1

Q
< Cw(x)I+E. 
(3) Fix SeQ and suppose w satisfies the reverse Holder inequality with exponent 1 + E. Then
. w(8) =
k (k XSW
<
w1+') 1/(1+,) 18 1'/(1+') < Cw(Q)
This gives (7.10) with 8 = £/(1
(I~\) ,/(1+,) o
+ E).
3. Al weights and an extrapolation theorem In this section we first give a constructive characterization of Al using the HardyLittlewood maximal function. This, combined with Proposition 7.2, lets us construct Ap weights for all p. We will then use this construction to prove a powerful extrapolation theorem: if an operator is bounded on L r (w) for a fixed index r and all weights w in A r , then for any p the operator is bounded on LP(w), w E Ap.
Theorem 7.7. (1) Let J. E Lfoc (IRn) be such that M f (x) < 00 a. e. If 0 < 8 < 1, then w (x) = M f (x)8 is an Al weight whose Al constant depends only on
8.
(2) Conversely, if wEAl then there exist f with
K,K I
E Lfoc(1Rn ),
8 < 1, and K
E L oo , such that w(x) = K(x)Mf(x)8.
Proof. (1) It will suffice to show that there exists a constant C such that for every f, every cube Q and almost every x E Q,
I~I
k
(Mf)'
< CMf(x)'.
Fix Q and decompose f as f = !1 + 12, where M!1(x) + Mh(x), and so for 0 < 8 < 1, Mf(x)8 < M!1(x/j
!1 =
fX2Q. Then Mf(x)
+ Mh(x)8.
Since M is weak (1,1), by Kolmogorov's inequality (Lemma 5.16)
<
3. Al weights and an extrapolation theorem
141
To estimate M 12, note that if y E Q and R is a cube such that y E R and 1121 > 0, then we must have that l(R) > ~l(Q), where l(·) denotes the side length of a cube. Hence, there exists a constant Cn, depending only on n, such that if x E Q then x E cnR. Therefore,
JR
1~lllhl < Ic:~1
LR 1121
< c~Mf(x),
and so M h(Y) < c~M f(x) for any y E Q. Thus
I~I ~ Mh(y)li dy < e:;liMf(x)li. (2) If w
E Al
then by the reverse Holder inequality there exists
such that
E
>
°
r 1+(0) 1/(1+10) < ~ r (~ IQI J w  IQI J w. Q
Q
This, together with the Al condition, implies that M(W 1+E)(x)I/(I+E) < CW(x) a.e. x E JR n . By the Lebesgue differentiation theorem (Corollary 2.13), the reverse inequality holds with constant 1, so if we let f = w 1+E and 8 = 1/(1 + E) we get
w(x) < M f(x)8 < Cw(x). The desired equality now follows if we let K(x)
= w(x)/Mf(x)8.
0
In part (1) of Theorem 7.7 we can replace f E Lfoc(JRn ) by a finite Borel measure p, such that Mp,(x) < 00 a.e. since the weak (1,1) inequality also holds for such measures. (This is readily shown using the covering lemmas described in Chapter 2, Section 8.6.) In particular, if 8 is the Dirac measure at the origin then M 8(x) = Clxl n ~ Thus Ixl a E Al if n < a < 0, and by Proposition 7.2, Ixla E Ap if n < a < n(p  I). This range is sharp since outside of it, Ixl a and Ixl a(l p') are not both locally integrable. Using Theorem 7.7 we can now prove our extrapolation theorem.
Theorem 7.8. Fix r, 1 < r < 00. 1fT is a bounded operator on Lr(w) for any wEAr, ·with operator norm depending only on the Ar constant of w, then T is bounded on LP(w), 1 < p < 00, for any w E Ap. Proof. We first show that if 1 < q < rand wEAl then T is bounded on Lq(w). By Theorem 7.7 the function (Mf)(rq)/(rI) is in Al since r  q < r 1, and by Proposition 7.2, w(M f)qr is an Ar weight. Therefore,
r
JlR
n
ITflqw =
r
JlR
n
ITflq(Mf)(rq)q/r(Mf)(rq)q/r w
7. Weighted Inequalities
142
the second inequality holds by our hypothesis on T and by Theorem 7.3 (since by Proposition 7.2, w E Aq), and the third inequality holds since If(x) I < M f(x) a.e. and q  r < 0, so M f(x)qr < U(x)lqr a.e. We will now show that given any p, 1 < p < 00, and q, 1 < q < min(p, r), T is bounded on LP(w) if w E A p/ q. The desired result follows at once from this: given w E A p , by Corollary 7.6 there exists q > 1 such that w E A p / q and so T is bounded on LP (w ) . Fix w E A p/ q. Then by duality there exists u E L(p/q)1 (w) with norm 1 such that
(in
ITfI Pw ) q/p =
L
ITflqwu.
For any s > 1, wu < M((wu)s)l/s and M((wuy)l/s E AI. Therefore, by the first part of the proof,
in
ITflqwu <
in
ITflqM((wu),)'/'
r IflqM((wu)S)I/s = c r Iflqw q/ pM((wuy)I/Sw q/ p J'R. <
c
J'R.n n
< C (/"" Ifl Pw ) q/p (/,;. M ((wu) ') (p/ q)' / 'w'(P/ q)') I/(P/q)' Since w E A p/ q, by Proposition 7.2, wl(p/q), E A(p/q)/. Therefore, if we take s sufficiently close to 1, wl(p/q)' E A(p/q)' Is. Hence, by Theorem 7.3 the second integral is bounded by
C
r (wu)(p/q)'wl(P/q), = c. J'R. n
This completes the proof.
o
A similar argument yields the following variation: given s > 1, if T is bounded on Lr(w) for all w E A r/ s , then for p > s it is bounded on LP(w) for all w E A p / s .
4. Weighted inequalities for singular integrals
143
4. Weighted inequalities for singular integrals In this section we prove weighted norm inequalities for Calder6nZygmund operators. Recall (see Definition 5.11) that a Calder6nZygmund operator is an operator which is bounded on L2, and for f E S can be represented by
T f(x)
=
J.
K(x, y)f(y) dy,
x
tf. supp(f),
JR."
where K is a standard kernel (i.e. one which satisfies (5.12), (5.13) and (5.14)). We begin with two lemmas. Lemma 7.9. 1fT is a Calder6nZygmund operator, then for each s
> 1,
M#(Tf)(x) < C s M(lfIS)(x)l/s, where M# is the sharp maximal operator (6.1). Proof. Fix s > 1. Given x and a cube Q containing it, by Proposition 6.5 it will suffice to find a constant a such that
I~I
k
al dy < CM(lfIS)(x)'/s
ITf(y) 
f
As in the proof of Theorem 7.7, decompose fX2Q. Now let a = T12(x); then
as
f = h + 12,
where
h =
,~,.Io ITf(y)  al dy
(7.12)
< Since s
> 1,
I~I
k
ITh(Y) I dy +
I~I
k
ITh(y) l'h(x)1 dy.
T is bounded on L S (JR n ). Therefore,
I~I kiTh(y)1 dy < (I~I
k
ITh(y)I SdY) lis
(1~11Q If(y)I' dY) 1/,
< 2n / sCM(lfI S)(x)1/s. We estimate the second half of the righthand side of (7.12) using (5.14); as before 1( Q) denotes the side length of Q.
I~I
k
ITh(Y)  Th(x) Idy
< IQ11
r r
JQ JJR.n\2Q
[K(y, z)  K(x, z)]J(z) dz dy
144
7. Weighted Inequalities
o <
Lemma 7.10. Let w E A p , 1 < Po < p
00.
If f is such that Mdf
E
LPO (w), then
r
J~n
IMdflPw
<
c
r
IM# flPw,
J~n where Md is the dyadic maximal operator (2.9) and M# is the sharp maximal operator (6.1), whenever the lefthand side is finite.
Proof. This is a weighted version of Lemma 6.9, and the proof is almost exactly the same. It will suffice to prove a good)" inequality which is a weighted analogue of Lemma 6.10: for some 6 > 0, w({x E JR~: Mdf(x)
> 2)..,M#f(x) < I')..}) <
CI'6 W ( {x E JRn : Mdf(x)
> )..}).
Since {x E JRn : Mdf(x) > )..} can be decomposed into disjoint dyadic cubes, it is enough to show that for each such cube Q,' w({x E Q: Mdf(x)
> 2)..,M#f(x) < I')..}) < CI'6 W (Q).
However, this is an' immediate consequence of inequality (6.6) (the same inequality with Lebesgue measure and with C",! on the righthand side) and the A= condition (7.10). 0 Theorem 7.11. If T is a Calder6nZygmund operator, then for any w E A p , 1 < p < 00, T is bounded on LP(w). Proof. Fix w E Ap. We may assume that f is a bounded function of compact support since the set of such functions is dense in LP (w). By Corollary 7.6, we can find s > 1 such that w E Apjs' Therefore, since Tf(x) < Md(Tf) (x) a.e., by Lemmas 7.9 and 7.10,
r
J~n
ITflPw
<
r
J~n
Md(Tj)P w
4. Weighted inequalities for singular integrals
145
r M#(Tf)Pw < 0 r M(IJIS)p/s w J~n < 0 r IflPw, J~n
<0
J~n
provided the second integral is finite. To show this it will suffice to show that Tf E LP(w). If the support of f is contained in B(O,R), then for E >
°
r
ITf(x)IPw(x) dx
J 1xl<2R
< (
r
w(x)I+E dX) 1/(l+E)
J 1xl<2R
r
(
ITf(x)IP(l+E)/E dX)
E/(l+E)
J 1xl<2R
By the reverse Holder inequality we can choose E such that the first integral is finite; the second is finite since T f E Lq, 1 < q < 00.
Ixl > 2R,
To complete our estimate, note that for
ITf(x)1
=
r f(y)K(x, y) dy < 0 J1yl
J~n
n
Therefore,
r ITf(x)IPw(x) dx < of r w(x) dx J 1xl>2R k=l J2 R
k
00
< 02:(2 kR)nPw(B(O, 2k+1 R)). k=1 Since w E A p , by Corollary 7.6 there exists q < p such that w E A q . Then by (7.3), which holds for balls as well as for cubes,
w(B(O, 2k+ 1 R)) < O(n, R, w)2knq. If we substitute this into the above expression we get a convergent series. Hence, Tf E LP(w) and our proof is complete. D Theorem 7.12. Let T be a Calder6nZygmund operator and let w E AI. Then
w({x
E
IRn : ITf(x)1 > A}) <
~
In
Iflw.
Proof. The proof is very similar to the proof of Theorem 5.1, the corresponding result without weights, and we use the same notation. Form the Calder6nZygmund decomposition of f at height A and decompose f
146
7. Weighted Inequalities
as f = g + b. It will suffice to estimate w( {x E IRn : ITg(x)1 > A}) and w({x E IRn : ITb(x)1 > A}). To estimate the first, note that by Proposition 7.2, w E A 2 . Hence, by Theorem 7.11
w( {x
IRn
E
:
ITg(x)1 > A}) <
<
:2 Ln
ITgl2w
~ Ln Igl2w 2n C
r
< ~ llRn Iglw. To complete this estimate we must show that J Iglw < C J Iflw. On the set IRn \ Uj Qj, g = fi on each Qj, since w E AI,
r Iglw < r IQ1.1 r lJ(y)1 dyw(x) dx = r If(y)lw(Qj) dy IQjl
lQj
lQj
J
lQj
lQj
r
f(y)w(y) dy.
lQj
For our second estimate, by (7.3) we have that
w(UQ;) < L w(Qi) < CL w(Qj) < CL ~6~1) IQjl· J
J
J
J
By our choice of the Qj's,
IQjl < ~
kj
Iii,
so by the same argument as above we can bound this sum. Now let Cj be the center of Qj. Then, since bj has zero average on Qj,
w( {x
E
lRn
\
UQi : ITb( x) I > ,\ }) J
<
~L j
=
~L j
l
ITbj(x)lw(x) dx
IRn\Qj
r
n
llR
*
r [K(x, y) 
K(x, cj)]bj(Y) dy w(x) dx.
\Q j lQj
By inequality (5.13), the last term is bounded by (7.13)
"""'1
C6 "\ /\ j
Qj
Ibj(y)1
(l
IRn\Qj
I Iy  Cjlc5 I +c5 w (x) dx ) dy. xCjn
5. Notes and further results
147
Arguing as in the proof of Lemma 7.9, we see that the term in parentheses is bounded by CMw(y); since w E AI, this in turn is bounded by Cw(y). Hence, (7.13) is bounded by
~
in
Iblw <
~
in
(If I + Igl)w <
~
in
Iflw,
where the last inequality follows from our previous argument.
o
Finally, recall that T is a Calder6nZygmund singular integral if it is a Calder6nZygmund operator such that T f (x)
(7.14)
= lim TE f (x) , E+O
where
TEf(x) =
r
K(x, y)f(y) dy. J1xyl>E (See Chapter 5, Section 4.) To determine when the limit (7.14) exists almost everywhere for f E LP (w), we need to consider weighted norm inequalities for the associated maximal operator, T* f(x)
= sup ITEf(x)l. E>O
Corollary 7.13. If T is a Calder6nZygmund operator, then for 1
T* is bounded on LP (w) if w E A p , and T* is weak (1, 1) with respect to w ifw E AI· 00,
Proof. Corollary 7.13 is a weighted version of Theorem 5.14 and its proof depends on Cotlar's inequality (5.18): for 0 < v < 1,
T* f(x) <
ev (M(ITfIV)(x)I/V + Mf(x)) .
When 1 < p < 00 the desired result follows at once from Theorems 7.3 and 7.11. When p = 1 the proof proceeds as in the proof of Theorem 5.14, given the following: Kolmogorov's inequality (Lemma 5.16) is true when Lebesgue measure is replaced by any positive Borel measure; Lemma 2.12 is true with Lebesgue measure replaced by w dx, when wEAl; and M is weak (1,1) with respect to w (by Theorem 7.1). 0
5. Notes and further results 5.1. References.
The Ap condition first appeared, in a somewhat different form, in a paper by M. Rosenblum (Summability of Fourier series in LP(/l) , Trans. ArneI'. Math. Soc. 105 (1962), 3242). The characterization of Ap when n = 1 is due to B. Muckenhoupt (Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207226). B. Muckenhoupt,
148
7. Weighted Inequalities
R. Hunt and R. Whee den (Weighted norm inequalities for the conjugate function and the Hilbert transform, Trans. Amer. Math.,Boc. 176 (1973), 227251) proved that w E Ap is necessary and sufficient for the Hilbert transform to be bounded on LP (w ). For p = 2 there is a different characterization due to H. Helson and G. Szego. (See Section 5.2 for more details.) R. Coifman and C. Fefferman (Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974),241250) extended these results to higher dimensions. (See Section 5.7 for more details.) Our proof is due to Journe [8]. In the same paper, Coifman and Fefferman also proved that A p ,weights satisfy the crucial reverse Holder inequality. This inequality first appeared in a paper by F. W. Gehring (The Vintegrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 34, 265277). The strong (p, p) norm inequality for the maximal function can be proved without the reverse Holder inequality: see the paper by M. Christ and R. Fefferman (A note on weighted norm inequalities for the H ardyLittlewood maximal operator, Proc. Amer. Math. Soc. 87 (1983), 447448). Also see Section 5.9 below. The characterization of Al weights in Theorem 7.7 is due to R. Coifman and R. Rochberg (Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249254). The extrapolation theorem is due to J. L. Rubio de Francia (Factorization theory and Ap weights, Amer. J. Math. 106 (1984), 533547), who proved it using the connection between weighted norm inequalities and vectorvalued inequalities; J. GardaCuerva (An extrapolation theorem in the theory of Apweights, Proc. Amer. Math. Soc. 87 (1983), 422426) gave a direct proof without using vectorvalued inequalities; our proof is simpler and seems to be new, at least in organization, although it follows the standard approach. The book by GardaCuerva and Rubio de Francia [6] contains an excellent exposition of the results related to weighted norm inequalities. See also the survey articles by B. Muckenhoupt (Weighted norm inequalities for classical operators, Harmonic Analysis in Euclidean Spaces, G. Weiss and S. Wainger, eds., vol. 1, pp. 6984, Proc. Sympos. Pure Math. 35, Amer. Math. Soc., Providence, 1979) and E. M. Dynkin and B. P. Osilenker (Weighted estimates for singular integrals and their applications, J. Soviet Math. 30 (1985), 20942154; translated from Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Moscow, 1983). 5.2. The HelsonSzego condition.
The first result characterizing weights for the Hilbert transform is due to H. Helson and G. Szego (A problem in prediction theory, Ann. Math. Pura Appl. 51 (1960), 107138): the Hilbert transform is bounded on L2(w) if and only if logw = u + Hv, where u,v E L OO and Ilvll oo < 7r/2. Their proof uses complex analysis; while it is straightforward to prove that their condition implies the A2 condition, no direct proof of the converse is known.
5. Notes and further results
149
The best that can be shown is that the above decomposition holds with Ilvll oo < 7r. This was first shown by R. Coifman, P. Jones and J. L. Rubio de Francia (Constructive decomposition of BMO functions and factorization of Ap weights, Proc. Amer. Math. Soc. 87 (1983), 675676). A higher dimensional analogue of this result was proved by J. Garnett and P. Jones (The distance in BMO to Loo, Ann. of Math. 108 (1978),373393). The HelsonSzego theorem condition has been generalized to LP (w) by M. Cotlar and C. Sadosky (On some LP versions of the HelsonSzego theorem, Conference on Harmonic Analysis in Honor of Antoni Zygmund, W. Beckner et al., eds., vol. 1, pp. 306317, Wadsworth, Belmont, 1983). 5.3. The Aoo condition.
As we already remarked, inequality (7.10) is called the Aoo condition SInce
This was proved independently by B. Muckenhoupt (The equivalence of two conditions for weight functions, Studia Math. 49 (1974), 101106) and by Coifman and Fefferman in the article cited above. We have already shown that Ap C A oo , so to prove this it will suffice to show that if w E Aoo there exists p such that w E Ap. The key step in the proof is to show that w1 satisfies a reverse Holder inequality with Lebesgue measure replaced by the measure w dx; that is, there exist positive constants E and C such that for every cube Q
(7.15)
1 ( w(Q)
r
i Q w1Ew
)
l/{l+E)
C
< w(Q)
r
i Q w 1 w.
This is equivalent to
I~I 10 w' < c (~~J, which is the Ap condition with
E
= p' 
1.
The proof of (7.15) is similar to the proof of Theorem 7.4. Starting from the Aoo condition we can prove a result analogous to Lemma 7.5 but interchanging the measures: there exist a' < 1, /3' < 1 such that if S. c Q and w(S) < a'w(Q) then lSI < /3'IQI. (Note that in the proof of Theorem 7.4 we did not use the full strength of Lemma 7.5 but only the existence of some a and /3.) To complete the proof, we use the fact that if w E Aoo then w is a doubling measure: there exists C > 0 such that for any cube Q, w(2Q) < Cw(Q). This property is sufficient to let us form the Calder6nZygmund
7. Weighted Inequalities
150
decomposition of a function f at height>" with respect to the measure w dx. This yields a collection of disjoint cubes such that
f(x) < >.. A<
w a.e. x
W(~j)
h,
rf.
UQj, j
fw
< CA.
The proof is essentially the same as the proof for Lebesgue measure (see Theorem 2.11). With this, we can complete the proof of (7.15) as in the proof of Theorem 7.4 by forming Calder6nZygmund decompositions of w I . If w E Aoo then by the above argument and by Proposition 7.2 there exists p > 1 such that for any q > p, w satisfies the Aq condition with uniform constant. Therefore, if we take the limit as q tends to infinity we get
(7.16)
C~I
hw)
< Cexp
C~I
h
logw) .
(Inequality (7.16) is called the reverse Jensen inequality.) Conversely, if w satisfies (7.16) for all cubes Q, then w E Aoo. This was proved independently by GardaCuerva and Rubio de Francia [6, p. 405] and S. V. Hruscev (A description of weights satisfying the Aoo condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (1984), 253257).
5.4. The necessity of the Ap condition for singular integrals. The Ap condition is necessary and sufficient for the HardyLittlewood maximal function to be bounded on LP (w ), 1 < p < 00, and to be weak (1,1) with respect to W, but we only showed th~t it is a sufficient condition for Calder6nZygmund operators. The Ap condition is also necessary in the following sense: if w is a weight on JRn such that each of the Riesz transforms is weak (p,p) with respect to w, 1 < p < 00, then w E Ap. In particular, the Hilbert transform is weak (p, p) with respect to w if and only if w E Ap. This is proved for the Hilbert transform in the article by Hunt, Muckenhoupt and Wheeden cited above, and their argument can be adapted to the case of Riesz transforms in JR n . Further, Stein [17, p. 210] has shown that if any of the Riesz transforms is bounded on LP(w), 1 < p < 00, then w E Ap.
5.5. Factorization of Ap weights. Proposition 7.2, part (3), lets us construct Ap weights from Al weights. It is a very deep fact that the converse is also true, so that w E Ap if and only if w = wow~P, where wo, WI E AI. This is referred to as the factorization theorem and was first conjectured by Muckenhoupt. It was proved by P. Jones (Factorization of Ap weights, Ann. of Math. 111 (1980),
5. Notes and further results
151
511530). A very simple proof was later given by R. Coifman, P. Jones and J. L. Rubio de Francia: see the article cited above or Stein [17, Chapter 5]. J. L. Rubio de Francia (in the paper cited above) studied factorization in a much more general setting and was able to prove a general factorization principle relating the factorization of operators with weighted inequalities. (Also see [6, Chapter 6].) The factorization theorem can be generalized to include information about the size of the exponent in the reverse Holder inequality. See the paper by D. CruzUribe and C. J. Neugebauer (The structure of the reverse Holder classes, Trans. Amer. Math. Soc. 347 (1995), 29412960). 5.6. Ap weights and BMO.
Let f be a locally integrable function such that exp(f) E A 2 . Given a cube Q, let fQ be the average of f on Q. Then the A2 condition implies that
(I~I fa eXP(f)) (I~I fa exp(f)) < C; equivalently,
C~I fa cxp(f 
fQ))
C~I fa cxp(fQ 
f)) <
0
By Jensen's inequality, each factor is at least 1 and at most C. Therefore, this inequality implies that
I~I
fa
cxp(lf  fQI) < 2C,
and so
I~I fa if 
fQI < 20
Hence, f E BMO. We have proved that if exp(f) E A2 then f E BMO; equivalently, if w E A2, logw E BMO. From the JohnNirenberg inequality one can prove that if f E BMO then for all ).. > 0 sufficiently small, exp()..f) E A2. (Cf. Corollary 6.13.) In other words,
BMO = {)"logw : )..
E
IR, wE A 2 }.
In fact, by Proposition 7.2 it follows that one can take A p , 1 place of A 2 .
00,
in
7. Weighted Inequalities
152
5.7. The CoifmanFefferman inequality. The original proof of Theorem 7.11 by Coifman and~Fefferman (see the reference given above) depended on the following inequality, which they proved for singular integrals T which satisfy the gradient condition (5.3): if W E Aoo then for 0 < p < 00, (7.17)
(T* is the maximal operator associated with T.) The heart of the proof is a goodA inequality relating M and T*: there exists 6 > 0 such that for all '"'( > 0,
W({x
E]Rn:
IT* f(x)1 > 2A,Mf(x) < '"'(A})
< C'"'(8 w ( {x
E ]Rn :
IT': f(x) I > A}).
R. Bagby and D. Kurtz (A rearranged good A inequality, Trans. Amer. Math. Soc. 293 (1986), 7181) gave another proof of (7.17) with an asymptotically sharp constant; they did so by replacing the goodA inequality with a rearrangement inequality:
(T* f)~(t) < C(M J)~(t/2)
+ (T f)~(2t),
where (g):O denotes the decreasing rearrangement of g with respect to the measure w dx. (Cf. Chapter 2, Section 8.2.) J. Alyarez and C. Perez (Estimates with Aoo weights for various singular integral operators, Boll. Unione Mat. Ital. (7) 8A (1994), 123133) extended inequality (7.17) to CalderonZygmund operators, but with T* replaced by T on the lefthand side. They did this by proving a sharper form of Lemma 7.9: for 0 < 6 < I,
M#(I T fI 8)(x)1/8 < C8 M f(x). Inequality (7.17) then follows from Lemma 7.10. 5.8. The strong maximal function. In Chapter 2, Section 8.8, we introduced the strong maximal operator Ms as the average of a function over rectangles with sides parallel to the coordinate axes. This operator satisfies weighted norm inequalities with weights analogous to Ap weights. The "strong" Ap condition is the following: W E A;, 1 < p < 00, if for any rectangle R with sides parallel to the coordinate axes,
Theorem 7.14. For 1 wE
A;.
00,
Ms is bounded on LP(w) if and only if
5. Notes and further results
153
Theorem 7.14 is a consequence of the corresponding result for the HardyLittlewood maximal function. For 1 < i < n, define Mi to be the maximal operator restricted to the ith variable: Md(XI, ... ,xn )
=
M f(xl, .. .
,XiI,·, Xi+I,·· .. ,
Xn)(Xi).
Then by Fubini's theorem, Msf(x) < (MI 0 .. · 0 Mn)f(x). By a limiting then for each i, W(XI' ... ,XiI,·, Xi+l, ... ,xn ) satisfies argument, if W E the onedimensional Ap condition with a uniform constant. Theorem 7.14 now follows at once from Theorem 7.3.
A;
If we define Ai = {w: Msw in terms of Ai weights. Theorem 7.15. Let 1 wQ, WI E
Ai
such that
W
<
p
<
< Cw a.e.}, then A; weights can be factored Then
00.
W
E A; if and only if there exist
= wQwi p .
This theorem was first proved by J.' L. Rubio de Francia; see the paper cited in Section 5.l. However, the analogue of Theorem 7.7 is false and (Msf)D, 0 < 0 < 1, need not be in Ai. This was proved by F. Soria (A remark on AIweights for the strong maximal function, Proc. Amer. Math. Soc. 100 (1987),4648). B. Jawerth (Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986), 361414) gave a different proof of Theorem 7.14 as a corollary to a much more general result. Let B be a basis, that is, a collection of open sets. Given a weight w, we define the weighted maximal function with respect to Band W by MB,wf(x) =
sup xEBEB
(IB) W
f
iB
Iflw
if X E UBEB Band 0 otherwise. If W = 1 we simply write MB. Note that if B is the collection of all rectangles R with sides parallel to the coordinate axes, then AIB = Ms. We say that a weight w satisfies the Ap condition with respect to B, 1 < p < 00, and write w E Ap,B, if for every B E B,
Theorem 7.16. Given a basis B, a weight wand p, 1 < p < wIpI. Then the following are equivalent:
00,
let cr =
(1) MB is bounded on both V(w) and V' (cr); (2) w E Ap,B, MB,w is bounded on Lpl (w) and MB,u is bounded on LP(cr).
The boundedness of MB,w and MB,u depends on the geometry of the basis B. If B is the basis of rectangles then the boundedness of these two operators was proved by R. Fefferman (Strong differentiation with respect
7. Weighted Inequalities
154
to measures, Amer. J. Math. 103 (1981), 3340) and by B. Jawerth and A. Torchinsky ( The strong maximal function with respect to measures, Studia Math. 80 (1984), 261285). For more general bases, ME,w is bounded on Lpl (w) if and only if the basis satisfies a covering property closely related to problems of differentiation of the integral. For further details see the article by Jawerth cited above. 5.9. Norm inequalities with two weights.
A very difficult problem is to generalize the results in this chapter to LP spaces with different weights; more precisely, given an operator T, to determine necessary and sufficient conditions on a pair of weights (u, v) for the operator to be bounded from LP(v) to LP(u), or to satisfy the weak (p,p) inequality
Complete answers to these questions are known for the HardyLittlewood maximal function. We generalize the Ap condition as follows: a pair of weights (u, v) is in A p , 1 < p < 00, if
and in Al if
Mu(x) < Cv(x)
a.e. x E
~n.
(When u = v this is exactly the Ap condition defined in Section 1.) With only minor changes we can adapt the proof of Theorem 7.1 to get the following result. Theorem 7.17. Given p, 1 < P <
00,
the weaktype inequality
u({xE~n:Mf(x»A})< ~
",P
r IflPv
IfR
n
holds if and only if (u, v) E Ap. For the strong (p, p) inequality, however, the Ap condition is necessary but is not sufficient: there exist pairs (u, v) in Ap for which the strong (p, p) inequality is false. This can be seen as follows: since the pair (u, M u) is in Al cAp, the pair ((Mu)lpl,u l  pl ) is in ApI (arguing as in the proof of Proposition 7.2, we have that (u,v) E Ap and (vlp',ulp') E ApI are equivalent conditions). Then the strong (p', p') inequality for the pair ((Mu)lp', u 1  p') with u = If I would imply that M is bounded on L1, a contradiction. (See Proposition 2.14.) Nevertheless, there is a characterization for the strong (p,p) inequality.
5. Notes and further results
155
Theorem 7.18. Given a pair of weights (u, v) and p, 1 following are equivalent:
< p <
00,
the
(1) M is a bounded operator from LP(v) to LP(u); (2) given any cube Q, (7.18)
10 M(v
1 p ' XQ)P u
10 v
1 p '
< 00.
Inequality (7.18) is equivalent to saying that M is bounded on the family of "test functions" v 1  p ' XQ. Theorem 7.18 was first proved by E. Sawyer (A characterization of a twoweight norm inequality for maximal operators, Studia Math. 75 (1982), 111). A simpler version of this proof was given by D. CruzUribe (New proofs of twoweight norm inequalities for the maximal operator, Georgian Math. J. 7 (2000), 3342). A very different proof was given by Jawerth in the article cited in the previous section. When u = v, (7.18) is equivalent to the Ap condition, so this result gives another proof of Theorem 7.3. This was proved by R. Hunt, D. Kurtz and C. J. Neugebauer (A note on the equivalence of Ap and Sawyer's condition for equal weights, Conference on Harmonic Analysis in Honor of Antoni Zygmund, W. Beckner et al., eds., vol. 1, pp. 156158, Wadsworth, Belmont, 1983). For singular integral operators, almost nothing is known except for the Hilbert transform. The Ap condition is necessary but not sufficient for the weak (p,p) inequality, 1 < p < 00; see the article by B. Muckenhoupt and R. Wheeden (Two weight function norm inequalities for the H ardyLittlewood maximal function and the Hilbert transform, Studia Math. 60 (1976), 279294). For p = 1, Muckenhoupt has conjectured that the Al ' condition is necessary and sufficient for the weak (1,1) inequality for the Hilbert transform. (Also see the next section.) A necessary and sufficient condition is known for the strong (2,2) inequality for the Hilbert transform, or, more precisely, for the conjugate function j on the unit circle. (See Chapter 3, Section 6.2.) This is due to M. Cotlar and C. Sadosky (On the HelsonSzego theorem and a related class of modified Toeplitz kernels, Harmonic Analysis in Euclidean Spaces, G. Weiss and S. Wainger, eds., vol. 1, pp. 383407, Proc. Sympos. Pure Math. 35, Amer. Math. Soc., Providence, 1979).
5.10. More on two weight inequalities. A general technique for applying weighted inequalities can be summarized as follows: given an operator T, first find a positive operator S such
156
7. Weighted Inequalities
that T is bounded from LP(Su) to LP(u). Then deduce boundedness properties for T from those of S. (We will use this technique in Chapter 8, Section 3 below.) To apply this technique we need to find such two weight inequalities. Theorem 2.16 above is a result of this kind: the HardyLittlewood maximal function is bounded from LP (M u) to LP (u). The first attempt to generalize this result to other operators was due to A. Cordoba and C. Fefferman (A weighted norm inequality for singular integrals, Studia Math. 57 (1976), 97101). They showed that if T is a singular integral satisfying the gradient condition (5.3), then T is a bounded map from LP(M(us)l/s) into LP(u). By Theorem 7.7, M(us)l/s E Al CAp, so this result now follows from Theorem 7.3. The next theorem gives a sharper result. Theorem 7.19. Let T be a singular integral satisfying the hypotheses of Theorem 5.1. Given a weight u, for 1 < p < 00,
r IT flPu < C JRnr Ifl PM[Pl+lu, p
JRn
where [P] is the integer part of p and Mk = Mo· .. 0 M is the kth iterate of the maximal operator. Further, this inequality is sharp since [P] + 1 cannot be replaced by [P].
Note in particular that singular integral operators are not in general bounded from LP(Mu) to LP(u). Since M(us)l/s E AI, Mku(x)
< Mk(M(uS)I/S)(x) < CkM(uS)(x)l/s,
so Theorem 7.19 is stronger than the result of Cordoba and Fefferman. It was first proved by J. M. Wilson (Weighted norm inequalities for the continuous square function, Trans. Amer. Math. Soc. 314 (1989), 661692) for 1 < p < 2, and for all p > 1 by C. Perez (Weighted norm inequalities for singular integral operators, J. London Math. Soc. 49 (1994), 296308). In the same paper Perez also proved the corresponding weak (1,1) inequality: u({x E IR n
:
ITf(x)1
>
A})
<
~
kn
If1M 2 u.
It is unknown if this inequality is true with M2 replaced with M.
Chapter 8
LittlewoodPaley Theory and Multipliers
1. Some vectorvalued inequalities Our approach to the study of LittlewoodPaley theory will be through the theory of vectorvalued singular integrals introduced in Chapter 5. In this section we prove some additional vectorvalued inequalities which we will need in later sections using those ideas as well as the weighted norm inequalities in Chapter 7. Theorem 8.1. 1fT is a convolution operator which is bounded on L2(JRn ) and whose associated kernel K satisfies the Hormander condition (5.2), then for any r, 1 < r < 00, and p, 1 < p < 00,
and for p = 1,
Proof. We use Theorem 5.17 with A = B = fT. The first inequality is immediate when p = r since T is bounded on LT(IRn), 1 < r < 00. Now consider the vectorvalued operator which associates to each sequence {fJ} the sequence {Tfj}. The kernel associated with this operator is K(x)l, where I is the identity operator on gT. For a convolution operator, conditions

157
8. LittlewoodPaley Theory and Multipliers
158
(5.20) and (5.21) reduce to
1
JJ(K(x  y)  K(x))IJJg7" dx < C, Ixl>2lyl which in turn is equivalent to the Hormander condition since JJ(K(x  y)  K(x))IJJg7"
= JK(x  y)  K(x)J. D
A natural problem is to generalize Theorem 8.1 to a sequence of operators {Tj} to get an inequality of the form
(8.1)
(~I1jfX )'/r p< Cp,r (~II;I' )'/r p
1 < p, r <
00.
It seems reasonable to conjecture that if the operators T j are uniformly bounded on L2 and the Hormander condition holds uniformly for their kernels K j then (8.1) holds. However, if p =1= r this can still be false. The vectorvalued operator which sends {fJ} to {TjfJ} has as kernel the operator in £(F,gr) which maps the sequence {Aj} to {Kj(X)Aj}. For this operator the Hormander condition becomes
1
sup JKj(x  y)  Kj(x)J dx < C, Ixl>21YI j and this condition is sufficient for (8.1) to hold. Nevertheless, we can use Theorem 8.1 to show that an inequality like (8.1) holds in particular cases. For example, we have the following result. Corollary 8.2. Let {Ij } be a sequence. of intervals on the real line, finite or infinite, and let {Sj} be the sequence of operators defined by (Sj (~) = Xlj(~)j(~). Then for 1 < r,p < 00,
ff
Proof. If I j
= (aj, bj)
then by (3.9) we have the formula '/,
Sjfj
=
2" (MajHMajfJ  MbjHMbjfJ) ,
with the obvious modifications if the interval is unbounded. The desired result now follows if we apply Theorem 8.1 to the Hilbert transform. D A common technique for proving vectorvalued inequalities like (8.1) involves weighted norm inequalities. The following result is an example of this approach.
2. LittlewoodPaley theory
159
Theorem 8.3. Let {Tj } be a sequence of linear operators which are bounded on L2 (w) for any w E A2 with constants that are uniform in j and which depend only on the A2 constant of w. Then for all p, 1 < p < 00,
(8.2)
Proof. Wheri p = 2 this result is immediate. Now suppose that p Then there exists a function u E L(p/2)' with norm 1 such that
> 2.
By Theorem 7.7, if 0 < 6 < 1 then M(u l / 8)8 is an Al weight (and so is in A 2 ) with a constant that depends only on 6. Since u(x) < M(u l / 8)(x)8 a.e., it follows that
Fix 6 such that 6(p/2)' > 1 and then apply Holder's inequality with exponents p/2 and (P/2)' to get inequality (8.2). Finally, if p < 2 then, since the adjoint operators Ty* are also bounded on L2(w), w E A2, we get (8.2) by duality. 0
2. LittlewoodPaley theory From the Plancherel theorem we know that if we multiply the Fourier transform of a function in L2 by a function of modulus 1, the result is again a function in L2. Similarly, if we multiply the terms of the Fourier series of an L2 function by ±1, we get another function in L2. However, neither of these properties holds for functions in LP, p f 2. Thus, whether a function is in LP depends on more than the size of its Fourier transform or Fourier coefficients. LittlewoodPaley theory provides a parti?l substitute in LP for the results derived from the Plancherel theorem. It shows that membership in LP is preserved if the Fourier transform or Fourier coefficients are modified by ±1 in dyadic blocks. For example, in the case of the Fourier series of an LP function, if we assign the same factor 1 or 1 to all the coefficients whose indices lie between 2k and 2k + l , we get the Fourier series of another LP function.
8. LittlewoodPaley Theory and Multipliers
160
We will first prove the corresponding result for the Fourier transform for functions defined on JR. Let Ilj = (2 j +1, 2j] U [2 j , 2j +1 ), and define the operator Sj by j E Z.
(The intervals [2 j , 2j + 1 ) are called dyadic intervals.) If f E L2 then by the Plancherel theorem
(L IS;!1
(8.3)
2 ) 1/2 2
=
1I!lb;
J
by the LittlewoodPaley theorem these two quantities are comparable in £P. Theorem 8.4 (LittlewoodPaley). Let f E LP(JR), 1
00.
Then there
exist positive constants cp and Cp such that
",,1I!lIp <
(8.4)
(L IS;!12)'/2
< Cpll!lIp_ P
J
We will prove Theorem 8.4 as' a consequence of a similar result, but one where the operators are defined using smooth functions instead of characteristic functions of intervals. Let 'ljJ E S(JR) be nonnegative, have support in 1/2 < I~I < 4 and be equal to 1 on 1 < I~I < 2. Define
'ljJj(~)
= 'ljJ(2j~)
(Sjfr(~)
and
It follows immediately that Sj Sj
=
= 'ljJj(~)j(~).
Sj.
The next result is the analogue of Theorem 8.4 for the S1's. We will use it to prove Theorem 8.4; it is also important in, its own right. Theorem 8.5. Let f E V(JR), 1 < p <
00.
Then there exists a constant
Cp such that
(8.5)
Proof. If ~ = 'ljJ and Wj(x) = 2j W(2 j x), then ~j = 'ljJj and Sjf = Wj * f. Therefore, given the vectorvalued operator which maps f to the sequence {Sjf}, it will suffice to prove that it is a bounded operator from LP to LP(f2). When p = 2, by the Plancherel theorem
(L IS;!1 J
since at any value of
f2 : l L 11j;;(~)12Ij(~W d~
2
=
< 311!1I1
J
~
at most 3 of the 'ljJ1's are nonzero.
161
2. LittlewoodPaley theory
To prove inequality (8.5) we need to show that the kernel {Wj} of the vectorvalued operator satisfies the Hormander condition; to do this it suffices to show that Ilwj(x)II£2 < Clxl 2 . (This follows from Proposition 5.2, which also holds in the vectorvalued case.) But
further, since W E S, Iw'(x)1 :::; Cmin(l, Ixl 3 ). Fix i so that 2 i < Ixl < 2i+1. Then the lefthand side is bounded by
C2::22j
+ Clxl 3 2:: 2 j < Clxl 2 . j>i
D
Therefore, by Theorem 8.5 we immediately get the righthand inequality in (8.4). From the polarization identity and (8.3) we get
Hence,
SllP{ fa jg : IIgllp' < 1} = SllP{ fa LA!· Sjg : Ilgll1" < 1}
11!llp =
J
<SUP{
(2)Sj!I't' J
< Cp
(L ISj!I') J
(2)Sjgl't' p
J
: Ilgllpl < pI
I}
1('
P
o These results can be extended to }Rn in two different ways. One analogue of Theorem 8.5 cuts the Fourier transform with smooth functions supported on the annuli 2j  1 < I~I < 2j +\ the other generalizes Theorem 8.4 with the cuts made by characteristic functions of products of dyadic intervals on the
8. LittlewoodPaley Theory and Multipliers
162
coordinate axes. We cannot generalize it by using characteristic functions of annuli: this follows from Theorem 8.32 in Section 8;3 below.
Theorem 8.6. Given 'IjJ E S(I~n) with 'IjJ(0) = 0, let Sj, j E Z, be the operator defined by (Sjfr(~) = 'IjJ(2j~)j(~). Then for 1 < p < 00,
t
(L:I Sj fI 2
(8.6)
J
Furthermore, if for all
~
2
i 0
L
(8.7)
1'IjJ(2j~)12
= C,
j
then we also have (8.8)
Ilfllp < C~
(L: ISjf12)
1/2
P
J
= 0,
Since 'IjJ E Sand 'IjJ(0)
L
it satisfies
1'IjJ(2 j 01 2 <
c.
j
Therefore, the proof of the first part of Theorem 8.6 is exactly the same as the proof of Theorem 8.5, while the proof of the second part is the same as the proof of the corresponding part of Theorem 8.4 since it follows from (8.7) that when p = 2 we have equality (up to a constant). There is a simple method for constructing functions which satisfy (8.7). Fix ¢ E S(n~n), nonnegative, radial, decreasing and such that ¢(~) = 1 if I~I < 1/2 and ¢(~) = 0 if I~I > 1. Then define , 'ljJ2(~) = ¢(~/2) _ ¢(~).
It follows immediately that
L
1'IjJ(2j~) 12
=
1,
~
i O.
j
We will state and prove the next result only for R,2; its extension to higher dimensions follows at once by induction. Dyadic rectangles in the plane are the Cartesian product of dyadic intervals in llt. The associated operators are then defined as the composition S] S~, where
(s]ff(6,6) = X~j(6)j(6,6) and
3. The Hormander multiplier theorem
163
Theorem 8.7. Let f E LP(IR 2 ) , 1 < p < constants cp and Cp such that
'1>lIfllp <
(I: IS}SUI
2)
),k
Then there exist positive
00.
1/2
< Cpllfllp" P
Proof. By the same argument as in the proof of Theorem 8.5, and with the same notation as there, we have that
(8.9)
(:~?~j1k12 )'/2
p
< Cp
(~11k12 )'/2
p
By Fubini's theorem, inequality (8.9) is true even if fk is a function of two variables and 8j acts only on one of them. Furthermore, arguing as in the proof of Theorem 8.4 (Corollary 8.2 readily adapts to this case) we can also replace 8j by 8 j since 8 j 8j = 8 j . But then we can first apply (8.9) and then Theorem 8.4 (along with Fubini's theorem since f is a function of two variables) to get
r
(f.IS} S'u1 2
2
p
< Cp
r
(~IS~fI2
2
p
<
C;llfllp D
3. The Hormander multiplier theorem VVe are now going to give some applications of LittlewoodPaley theory. In the next three sections we will consider the problem of characterizing multipliers on LP. More precisely, given a function m, when is the operator Tm defined by (TmJ)~ = mj bounded on LP, 1 < P < oo? (We first considered multipliers on LP(IR) in Chapter 3, Section 5; also see Chapter 3, Section 6.7.) The Sobolev space L;(IRn) is defined to be the set of functions 9 such that (1 + 1~12)a/2g(~) E L2 and the norm of this function is the norm of 9 in (When a is a positive integer this definition is equivalent to that given in Chapter 4, Section 7.7.) Note that if a' < a, L;(IRn) c L~/(IRn).
L;.
Proposition 8.8. If a > n/2 and 9 E L;(IRn) then is continuous and bounded.
9E
L1; in particular, 9
D
8. LittlewoodPaley Theory and Multipliers
164
It follows from this result that if m E L~ with a > n/2 then m is a multiplier on LP, 1 < P < 00. In fact, if (T = mj then T f = K * f with K E L1. Hormander's theorem shows that m is a multiplier on LP under much weaker hypotheses. To prove it we first need to prove a weighted norm inequality.
ff
Lemma 8.9. Let m E L~, a > n/2, and let>. by (T)..ff(~) = m(>.~)j(~). Then
> O. Define the operator T)..
r IT)..f(x)1 2u(x) dx < C r If(x)12 Mu(x) dx, J~n J~n where the constant C is independent of u and >., and M is the HardyLittlewood maximal operator. Proof. If k = m then by our hypothesis, (1 and the kernel of T).. is >. n K (>. 1 x). Hence,
1 ~n
2
+ IxI 2)a/2 K(x)
= R(x) E L2,
11
>.nR(>.l(x  y)) 2 f (y)d y u(x)dx ( 1 )1 )a/2 1( 2 ~n ~n 1+>' xy >. nlf(y)1 2 < IImlli~ J~n J~n (1 + 1>.l(x _ y)1 2)a u(x) dydx
IT)..flu=
r r
< Callmllh
Ln If(y)1 2Mu(y) dYi
the second inequality follows by applying the CauchySchwarz inequality and using the fact that IIRII2 = IImIIL2, and the third inequality follows if a we integrate in x because (1 + IxI 2)a is radial, decreasing and integrable since a > n/2. (See Proposition 2.7 and Theorem 2.16.) 0 To state Hormander's theorem, let 'If; E C= be a radial function supported on 1/2 < I~I < 2 and such that
=
2::=
1'lf;(2j~) 12
=
1,
~
# O.
j==
Theorem 8.10 (Hormander). Let m be such that for some a
sup Ilm(2 j ·)'lf;II£2a .
> n/2,
< 00.
J
Then the operator T associated with the multiplier m is bounded on 1 < p < 00.
LP(l~n),
Proof. We apply Theorem 8.6. First, define the family of operators {Sj} by (Sjff(~) = 'If;(2j~)j(~). Then inequality (8.8) holds by our choice of'lf;. Now let ;j; be another C= function supported on 1/4 < I~I < 4 and equal
3. The Hormander multiplier theorem
165
to 1 on 1/2 < I~I < 2. If we define Sj by (Sjfr(~) = (fi(2j~)j(~), then SjSj = Sj and the family {Sj} satisfies inequality (8.6). (Note that (8.7) does not necessarily hold for (fi.) Therefore,
The multiplier associated with SjT is 'lj;(2j~)m(~), so by our hypothesis and by Lemma 8.9,
where C is independent of j. Using this inequality, by an argument analogous to that in the proof of Theorem 8.3 we can prove that for p > 2
(~ISjT9jI2)'/2
(~19jI2)'/2
•
p'
Therefore, if we combine this with inequality (8.6) for {,gj} we get
IIT!II. < G
(L ISj!1
< GII!llp,
2 ) 1/2
p> 2.
P
J
Finally, for p < 2 this inequality follows by duality. (See Chapter 3, Section 6.7.) D Hormander's theorem is usually stated in the following way. Corollary 8.11. If for k = [n/2] for 1111 < k
(8.10)
(~
sup Rim
+ 1, m
f
R JR
R
then m is a multiplier on LP, 1 < p <
E C k away from the origin, and if
ID/3m(~)12 d~) 00.
1/2
< 00,
In particular, m is a multiplier if
Proof. If we make the change of variables ~ ft R~ in inequality (8.10) and use the fact that D/3m(R·)(~) = RI/3I(D/3m)(R~), we get
sup R
(
I,
1<1~1<2
ID/3m(R·)(~)12 d~
1/2 )
< C.
8. LittlewoodPaley Theory and Multipliers
166
Since D(3(m(2j.)'I/J)(~)
=
2:
C'Y,(3D'Ym(2j.)D(3~'Y'I/J,
bl s:: 1(31 and since ID!1'I/J1 :::; c, it follows that sup IIm(2j')'l/JIIL2k apply Theorem 8.10.
<
00
and so we can D
Example 8.12.
(1) m(~) = 1~lit satisfies the last hypothesis of Corollary 8.11. In fact, it is easy to see that ID!1m(~)1 < Ctl~II!1I. (2) If m is homogeneous of degree 0 and of class C k on the unit sphere for some k > [n/2] then m is a multiplier on LP(JRn ), 1 < p < 00.
4. The Marcinkiewicz multiplier theorem By Corollary 3.8 we know that if m is of bounded variation on JR then it is a multiplier on LP. The Marcinkiewicz multiplier theorem shows that we can weaken this hypothesis to a condition on dyadic intervals. Theorem 8.13. Let m be a bounded function which has uniformly bounded variation on each dyadic interval in JR. Then m is a multiplier on LP(JR), 1 < p < 00. Proof. The proof is similar to the proof of Corollary 3.8. Given a dyadic interval I j , let Tj be the operator associated with the multiplier mXlj' We will consider the case I j = (2 j , 2j + 1 ); the other case is handled in exactly the same way. For ~ E I j
(mXIJ(~) = m(2j) + .
(E", am(t); ~2J
without loss of generality we may assume that m is right continuous on JR \ {o}. But then 2H1
Tjf(x) = m(2 j )Sjf(x)
+ l
~2J
St,2H1 f(x) dm(t),
where St,2 j +1 is the operator associated with the multiplier X[t,2H1]. By (3.9) and Theorem 7.11, the operators 8j and St,2H1 are uniformly bounded on L2(w) if wE A 2. Therefore, by Minkowski's inequality and our hypotheses, the operators T j are also bounded on L2 (w) with a constant depending only on the L= norm of m and the total variation of m on I j . Hence, by Theorems 8.4 and 8.3, if T is the operator associated with m then
167
4. The Marcinkiewicz multiplier theorem
=
(L ITjSj!12 )'/2
C
P
J
(LISj!12f2
P
J
o We can extend Theorem 8.13 to higher dimensions; we first do so in
]R2.
Theorem 8.14. Suppose m is a bounded function in the plane, twice differentiable in each quadrant of JR2 and such that
am sup /, a (tl, t2) dtl j Ij tl
< 00,
am a (tl, t2) dt2 < 00, J Ij t2 a2m sup a a (tl, t2) dtldt2 < 00, i,j Ii xlj tl t2 s~p /,
1
where Ii and I j are dyadic intervals in JR. Then m is a multiplier on V(]R2), 1 < p < 00. Proof. As before, we restrict our attention to dyadic intervals in JR+. Let Ii = (2i,2i+1), I j = (2 j ,2 j +I ), and fix (6,6) E Ii X I j . Then
m(6,6)
=
a2m 12i61~2 a a (tl, t2) dtldt2 2j tl t2
1
6 am a (iI, 2J) dtl 2i tl + m(2i, 2j ).
+
0
+
16am a (2\ t2) dt2 2j t2 0
If Ti,j is the operator associated with the multiplier mXli xlj then
Ti,jf(x)
!
82m
I
2
= IixIj a tl a t2 (tl' t2)St 1, 2i+ 1St2, 2i+1f(x) dtldt2
+ }Ii r
aam (tl' 2j )Si 2i+1f(x) dtl tl 1,
+ }Ir j
aaom (2i, t2)S; 2j+1f(x) dt2 t2 2,
+ m(2i, 2j )SlsJ f(x), where the superscripts denote the variable on which the operator is acting. Therefore, Ti,j is bounded on L2 (w) where w is any weight which satisfies the A2 condition uniformly in each variable (i.e. if w E A 2; see Chapter 7,
8. LittlewoodPaley Theory and Multipliers
168
Section 5.8). We can now argue as we did in the proof of Theorem 8.13 to get the desired result. 0 The proof of Theorem 8.14 extends immediately to higher dimensions; the only problem is that the notation quickly becomes unwieldy. In ~n the hypotheses become . sup. ]1, ...
,]k
J
[)km
[) .... ~21
Ih x ... xljk
a~2k. (~)
d~il··· d~ik
< 00,
where the Ij's are dyadic intervals in JR and the set {il, ... ,ik} runs over all the subsets of {I, ... ,n} containing k elements, 1 < k < n.
5. BochnerRiesz multipliers As we discussed in Chapter 1, Section 9, the spherical convergence of Fourier integrals consists in studying when
SRf(x)
=
r
Jf.I5: R
j(~)e27rif.'x d~
1
converges to f (either in LP norm or pointwise almost everywhere) as R for f E £P. The multiplier of this partial sum operator is X{If.I5:Rf
+ 00
(SRfr(~) = X{If.I5:R}j(~). When n = 1 we already showed (Proposition 3.6) that this multiplier is bounded on £P(JR), 1 < p < 00. For n > 2 this multiplier is not well behaved; see Section 8.3 below. In higher dimensions, the problem is simpler if we consider multipliers which are more regular than the characteristic. function of a ball. For example, if we take the Cesaro means of the operators St for o < t < R, we get the operator '
oRf(x) =
~ foR Sd(x) dt = ~bR (1 ~ I~) j(Oe2ni,'x d€.
This example leads us to consider the following family of operators:
(S'RfnO
=
(1 ~ ~): j(€),
a>
0,
where A+ = max(A, 0). For a fixed value of a the operators {SjU are all dilations of the operator S1' so to prove that these operators are uniformly bounded on LP it suffices to consider" the case R = 1. When n = 1, the Fourier transform of the function m(~) = (1  I~I)+ is in LI. (This can be shown directly by applying integration by parts and then arguing as in the proof of the RiemannLebesgue lemma (Lemma 1.4) to show that Im(x)1 < C(1 + Ixl)la.) Therefore, by Young's inequality (Corollary 1.21)
5. BochnerRiesz multipliers
169
m is a multiplier on LP, 1 < P < no longer true.
00.
However, in higher dimensions this is
Rather than considering these operators directly, it is customary to replace them with the BochnerRiesz multipliers:
(T a fr(~)
(1  1~12)~j(~),
=
a
>
o.
For if 'ljJl and 'ljJ2 are C= functions on IR with compact support such that 'ljJl(I~I) = 'ljJ2(1~1) = 1 if I~I < 1, then we can write (1  1~12)~
+ 1~I)a'ljJl (I~I)]' (1 _1~12)~[(1 + 1~1)a'ljJ2(1~1)]·
=
(1 I~I)~ =
(1  I~I)~ [(1
Since each term in square brackets is a Hormander multiplier, it follows that T a is a bounded operator on some LP, 1 < p < 00, if and only if Sf is. Our main result is the following. Theorem 8.15. The BochnerRiesz multipliers T a satisfy the following:
nl (1) If a > 2 then T a is bounded on LP(IRn ), 1
00.
nl (2) If 0 < a < 2 then T a is bounded on LP(IRn ) if 1
a
p
1 2
nl'
1
1
2a + 1
<
and is not bounded if
2 > P 2n . For the values of p not considered in Theorem 8.15, see Section 8.3 below. The value (n  1) /2 is called the critical index. The singularity of the BochnerRiesz multipliers is on the circle I~I = 1; therefore, to prove Theorem 8.15 we are going to decompose the multipliers on dyadic annuli whose widths are approximately their distances to the unit sphere. To be precise: choose functions ¢k E C=(IR) which are supported on 1  2 k + 1 < t < 1  2 k  1 and are such that 0 < ¢k < 1, ID,B¢kl < C,B2 kl ,B1 for all multiindices {3 (where C,B is independent of k), and for 1/2 < t < 1
=
L ¢k(t)
=
1.
k=l
Now define
= ¢o(t)
= 1 L¢dt), k=l
0
< 1/2,
8. LittlewoodPaley Theory and Multipliers
170
and let ¢o(t)
= 0 if t
~
1/2. Then 00
(1  1~12)~ =
L(1 1~12)a¢k(I~I)· k=O
On the support of ¢k the size of 1  1~12 is approximately 2 k , so we define
in order to write 00
(1  1~12)~ =
L
2ka¢k(I~I)·
k=O
This allows us to decompose the operator T a as 00
T af =
L 2 kaTkf, k=O
where
The behavior of operators like the Tk's is given by the next lemma, which will yield the positive part of Theorem 8.15. ~
Lemma 8.16. Given 0 < 8 < 1, let ¢ be a function on IR which is supported
on 1  48 < t < 1  8 and is such that 0 < ¢ < 1 and ID,6¢1 < 081,61 for any fJ. Then for any E > 0 the operator To associated with the multiplier ¢(I~I) satisfies
(8.11)
Proof. Fix K so that k(~) = ¢(I~I) and let a be a positive even integer. Then by inequality (1.20) and the Plancherel theorem,
II (1 + Ixl a)KII2 < Oil (I + (_~)a/2)¢II2 < 08 1/ 2(1 + 8 a) < 08 a+1/ 2; the second inequality follows by our hypotheses on the support of ¢ and the size of Da¢. By Holder's inequality this holds for arbitrary a > O. Given a, fix s > 1 such that as is an even integer. Then
11(1 + Ixl a)KII2 < 011(1 + Ixl as )1/sKII2 < 11(1 + Ixlas)KII~/sIIKII~/sl, and the desired inequality now follows from the previous argument. Now let a
= n/2 + E. Then by Holder's inequality
IIKll1 <
11(1 + Ixla)KII211(1 + Ixl a)1112 < OE8(n;1+E).
171
5. BochnerRiesz multipliers
Therefore, by Young's inequality (Corollary 1.21) with q = 1,00,
If we interpolate between these two inequalities and the trivial inequality when q = 2 we get (8.11). 0 To prove the negative part of Theorem 8.15 we need two lemmas.
Lemma 8.17. Ifm is a function with compact support which is a multiplier on LP for some p, then m E £P. Proof. Let f E S be such that and so Tmf E LP, but (Tmff =
j = 1 on mj = m.
the support of m. Then
f
E £P
0
Lemma 8.18. The Fourier transform. of (1  1~12)+ is
Ka(x) = 7r ar(a + 1)lxl~aJ!!+a(27rlxl), 2
(8.12)
where JJ.t is the Bessel function JJ.t(t)
=
GjJ.t
111
r (p, + '2) r ('2)
eits (1_
S2)J.t~ ds.
1
Proof. Inequality (8.12) follows immediately from two results: (1) If f(x)
= fo(lxl)
E L1(I~n) is radial then
j(~) = 27r1~11~ (2) If j
j
is radial and
r= fo(s )J!!l (27r1~ls )sn/2 ds.
io
2
> 1/2, k > 1 and t > 0 then tk+1 (I Jj+k+1(t)
=
2kr(k + 1) io Jj(ts)sj+1(1  s2)k ds.
The proofs of these can be found in Stein and Weiss [18, pp. 155, 170].
Proof of Theorem 8.15. By Lemma 8.16, for each k
IITkfilp <
CE2k(n;1+E)I~111Ifllp.
Therefore, by Minkowski's inequality
IITa flip < C
E
f
2k( n;l +E) I~llka Ilfll p.
k=O
The positive results now follow immediately.
0
8. LittlewoodPaley Theory and Multipliers
172
To prove the negative results, note that from the definition of the Bessel function we have that J/L(t) = O(t/L) as t + o. Further, as t + 00, J/L(t) ~ t 1 / 2 (see Stein [17, p. 338]). Hence, by Lemma 8.18
and
Therefore, K a E LP only if
2n p> n+ 1 +2a' and so if we combine Lemma 8.17 with a duality argument (see Chapter 3, Section 6.7) we get that T a cannot be bounded if 1
2a + 1
1
>P 2 2n .
o 6. Return to singular integrals In this section we apply LittlewoodPaley theory to singular integral operators. Our main result is the following. Theorem 8.19. Let K E Ll(JRn ) be a function of compact support such that for some a > 0
and such that
r K(x) dx = o.
I~n For each integer j let Kj(x)
= 2 jn K(2 j x). Then the operator 00
Tf(x)
=
L j=oo
satisfies
Kj
* f(x)
6. Return to singular integrals
173
The L2 norm inequality for the operator T follows at once from the Plancherel theorem and our hypotheses. Since K has compact support we also have that JK(~)J < CJ~J, so 2
fan JTf(x)J2 dx = fan ~ Kj(~) Jj(~)J2 d~ J
<
fa. (~IK(2jell) 2lj(Ol2de
< C IrK'
(~min(l2j€la, 12j W)' Ij({lI2~
< CllfIJ~·
> 1 we will decompose the oper
To prove Theorem 8.19 for arbitrary p ator as
where each Tk is gotten by cutting up each multiplier Kj into pieces supported on the annuli 2 j  k  1 < J~J < 2 j  k +1 (which we will denote by I~I ~ 2 j  k ) and summing in j. For each k the corresponding pieces have supports which are almost disjoint and whose size depends only on k. Because of this, the sum of the norms of the Tk's will be dominated by a convergent geometric series.
Proof. Let 'ljJ E S(JR n ) be a radial function whose Fourier transform is supported on 1/2 < J~J < 2 and which satisfies 00
L
;j;(2k~) = 1,
~
==f.
o.
k=oo
Let 'ljJk(X)
= 2 kn 'ljJ(2 kx)i
then by the Plancherel theorem 00
Kj =
L
Kj
* 'ljJj+k.
k=oo
Now define 00
00
j=oo
j=oo
8. LittlewoodPaley Theory and Multipliers
174
Then we have that
* f = LKj * 'l/Jj+k * f = L'ilf.
Tf = LKj
j,k
j
k
Suppose the following inequalities were true: IITkfl12 < C2 a1k1 1IfI12,
(8.13) (8.14)
I{x E
~n : ITkf(x)1 > ).}I <
C(l; Ikl) Ilfll1.
(In fact, we will show (8.13) provided 0 < a < 1. If a> 1 then the constant will be 2 lkl but this will not affect the proof.) If we apply the Marcinkiewicz interpolation theorem (cf. (2.2)) to (8.13) and (8.14), then for 1 < p < 2 we get that
IITkfilp < Cp2aOlkl(1 + Ikl)lollfll p, where lip = 8/2 + (1  8). The desired result follows if we sum in k for 1 < p < 2; by a duality argument we get it for 2 < p < 00. It remains, therefore, to prove (8.13) and (8.14). The first follows from the definition: since ~j'(j;m is nonzero only if m = j, j ± 1, by the Plancherel theorem j+l
IITkfll~ < L
L
j m=jl
<
i
Ikj(~)km(~)II~j+k(~)~m+k(~)llj(~)12 d~
]Rn
CI: It,I~2;k T2alkllj(OI2~ J
< C22alklllfll~· The second inequality follows since Ikj(~)1
< rnin(12j~la, 12j~l).
To prove (8.14) we will apply Theorem 5.1; it will suffice to show that the constant in the Hormander condition (5.2) for Tk is C(l + Ikl). Fix y # 0; then
r
J1xl>2lyl
f
j=oo
00
(K * 'l/Jk)j(X  y)  (K * 'l/Jk)j(X) dx
r
2 jn l(K * 'l/Jk)(2 j x  2 j y)  (K * 'l/Jk)(2 j x) 1 dx j=oo J1xl>2lyl .
< L
=L
r . J1xl>2
J
= LIj, j
1 j
lyl
I(K * 'l/Jk)(X  2 j y)  (K * 'l/Jk)(x)1 dx
6. Return to singular integrals
175
where 1j denotes the jth summand. If we make the change of variables y ~ 2y then 1j becomes 1j 1. Since this does not affect the sum, we may assume that 1 < Iyl < 2. We have several different estimates for the 11's. First, by the mean value theorem, 1j
r r IK(z)"~k(X  2 y  z)  'lfJk(x  z)1 dzdx < r IK(z)1 r l'1fJk(X  2 y)  'lfJk(x) I dx dz JIRn JIRn <
j
JIRn JIRn
j
< CIIKII 1 2 j 
k.
We will use this estimate if k + j > 0; if k inequality we get the better estimate 1j < C.
+j <
0 then from the first
To get the third estimate we may assume without loss of generality that the support of K is contained in B(O, 1). But then 1j
<
21
<
2
Ixl>2 j
r Jz l50 1
I(K * '1fJk)(X) I dx
IK(z) I 1
r Jxl>21
1'1fJ(x  2 kz) Idx dz. j  k
If j < 0 then Ixl > 212 kzl, so that Ix  2 kzl > Ixl/2. Hence, 1j
.
< 211Klh
Jrxl>2 .
1'1fJ(x) I dx < C2 j +k.
J k 1
1
From these estimates we get the following: if k > 0,
if k
< 0,
o
This completes the proof.
Theorem 8.19 has several applications. The first is a simpler proof of Theorem 4.12; it requires the following lemma.
Lemma 8.20.
lfn
E Lq(sn1), q> 1, and
K(x)
=
D(x') W X{l
8. LittlewoodPaley Theory and Multipliers
176
then for a <
1/ q'
Proof. If we rewrite the Fourier transform in polar coordinates we get
k(~) =
r O(u) 12 e 27fir (u.c,)drr d(J"(u) = }snl r O(u)I(u·~)d(J"(u), }snl 1
where
I(t)
=
12
e27firtdr. r
1
Since II(t)1 Hence,
< min(I,ltl I ), we have that II(t)1 < Itl a ,
°<
a < l.
Ik(~)1 < I~Ia !snl 10(u)llu· ~'Ia d(J"(u) I/q' ) aq < lelallnll (!sn, lu· el ' do"(u) The last integral does not depend on e and is convergent if aq' < 1. q
Corollary 8.21. If 0 E Lq(snI), q> 1, and JO(u)d(J"(u) singular integral
Tf(x) is bounded on LP, 1 < p <
= p. v.
r
O(y') JlR lYF f(x n
0
= 0, then the
y) dy
00.
If we define K as in Lemma 8.20 then 00
Tf=
L
Kj*f
j=oo
and the desired result follows immediately from Theorem 8.19. Our second application is to a family of square functions. Corollary 8.22. If K and K j are defined as in Theorem 8.19 then the square function
g(f) = is bounded on LP, 1 < p <
00.
(
jJ;= IKj* II'
1/2 )
6. Return to singular integrals
177
Proof. First note that given a sequence Ej = ±1, if we define the operator
E
=
{Ej}
such that for each j,
TEf = LEjKj * f, j
then it follows from the proof of Theorem 8.19 that the constant Cp independent of E.
IITEfllp < Cpllfllp
with
Recall that the Rademacher functions are defined as follows: ro () t
={
I,
1,
0
and for j > 1, rj(t) = ro(2jt). (We extend ro to all of IR as a periodic function.) The Rademacher functions form an orthonormal system in L2 ([0, 1]); furthermore, if 00
L ajrj(t) E L2([0, 1])
F(t) =
j=O
then F E LP([O, 1]), 1 < p < Bp such that
and there exist positive constants Ap and
00,
ApllFllp < 11F112 =
(~lajI2) 1/2 < BpllFllp
(See Stein [15, p. 104].) Therefore,
g(f)(x)"
=
(L IKj * f(x)1
2 ) p/2
J
<
B~ [
L Tj(t)Kj * f(x) Pdt. J
If we integrate with respect to x and apply Fubini's theorem, then for each t we have an operator like T E , so the desired inequality follows by the above observation. 0
For our final application, note that in the statement of Theorem 8.19 and Corollary 8.22 we can replace K E L1 by a finite Borel measure with compact support whose Fourier transform decays quickly enough. Corollary 8.23~ If f.J, is a finite Borel measure with compact support and such that for some a > 0,
then the maximal function Mf(x) is bounded on LP, 1
= s~p 00.
in
f(x  2jy) df.J,(y)
8. LittlewoodPaley Theory and Multipliers
178
Proof. Let ¢ E S(IRn) have compact support and be such that <;6(0) = l. Define the measure a = pjl(O)¢; then a satisfies the hypotheses of Theorem 8.19. Therefore, from the definition of M and from Proposition 2.7 we have that
Mf(x) <
<
s~p laj * f(x)1 + Ijl(O)1 s~p
(~IUj * f(X)1
2 ) 1/2
Ln
f(x  2jy)¢(y) dy
+ IP(O)IMf(x),
where M is the HardyLittlewood maximal operator. By Corollary 8.22 the square function in the last term is bounded on £P, so we are done. D If p is Lebesgue measure on the unit sphere
(
JlR
snl
then the integr~l
f(x  2jy) dp(y)
n
is the average of f on the sphere centered at x with radius 2 j . In this case the corresponding maximal function is called the dyadic spherical maximal function. Since Ijl(~)1
<
CI~I(n+l)/2
(apply Lemma 8.18 with fa replaced by the Dirac delta at 1), Corollary 8.23 immediately implies the following result. Corollary ,8.24. If n > 2 then the dyadic spherical maximal function is bounded on LP, 1 < p < 00.
7. The maximal function and the Hilbert transform along a parabola Let L be the parabolic operator
8u 82u Lu = 8X2  8xI. If we take the Fourier transform then, arguing as in Chapter 4, Section 5, we get
where
7. The Hilbert transform along a parabola
179
Unlike the multipliers we considered in Chapter 4, these multipliers are not homogeneous of degree o. However, they do satisfy the following condition:
By an argument analogous to the proof of Proposition 4.3, we can show that if K(Xl' X2) is the Fourier transform of m then K satisfies the homogeneity condition
This example leads us to consider operators of the following ~ype. Given (Xl, X2) E JR2 \ {(O, On, there exist unique values r > 0 and 8 E Si such that Xl = r cos( 8) and X2 = r2 sin( 8). Define 0(8) = K( cos( 8), sin( 8)); then K(xl, X2) = r 3 0(8). Define the operator T by
Tf(xl, X2) = p. v.
=
p. v.
r K(Yl, Y2)f(Xl 
J~2
Yl, X2  Y2) dy l dY2
0(8)  r cos(8), X2  r2 sin(8))(1 + sin (8)) drd8. 1o=127r f(Xl r 2
0
If 0(8 + 7r) = 0(8) (i.e. if K is odd), then by the same argument as we used in the method of rotations (see Corollary 4.8),
Tf(Xl,X2) =
10
7r
0(8)He(Xl,X2)(1+sin2(8))d8,
where
Hef(Xl, X2)
=
p. v.
rf(Xl  r cos(8), X2  r2 sgn(r) sin(8)) dr.
J~
r
The operator He is the Hilbert transform along the (odd) parabola (rcos(8),r 2 sgn(r)sin(8)), r E JR. (When 8 = 0 or 7r/2 this is a line and we get the usual Hilbert transform.) By the proof of Proposition 4.6, if o E Ll(8 l ) and if the operators He were uniformly bounded on LP, then T would also be bounded. For simplicity, the remainder of this section is devoted to proving that the Hilbert transform and the maximal function along the parabola r(t) = (t, t 2 ) are bounded on LP(lR 2 ), 1 < p < 00. More precisely, we will consider
Hr(Xl, X2) = p. v.
r f(Xl  t, X2 _ t 2) dt, J~ t
Mr(Xl, X2) = sup Ih h>O
2
jh f(Xl  t, X2  t 2) dt h
A similar argument shows that the operators He are uniformly bounded.
180
8. LittlewoodPaley Theory and Multipliers
Our approach will be to decompose the operators Hr and Mr in a way similar to the decomposition done in the previous section. For j E Z, let o"j and JLj be finite measures which act on a continuous function g by
0"j (g) =
(8.15)
1
2
2j
(8.16)
JLj(g)
1
1 = ~ 2J
dt
g (t, t ) , t g(t, t 2 ) dt.
2j
Then 00
(8.17) 00
and
(8.18)
< 2 sup JLj * If I;
Mr f
j
the latter inequality holds since if 2j < h < 2j + 1 then
j 2h 1
h
h
f(xl  t, X2
1 t ) dt < 2j+l
j
,L
2

HI
2
JLi
* If I·
z=oo
Using these decompositions, we will derive the boundedness of Hr and Mr from two more general results. Hereafter, given a measure 0", let 10"1 . denote its total variation measure and let 110"11 denote its total variation. Theorem 8.25. Let {O"j }jo: be a sequence of finite Borel measures with
II 0" j" < C and such that for some a > 0 (8.19)
If the operator
0"* (1)
=
sup IIO"j I * fl j
is a bounded operator on Lq for some q > 1) then the operators 00
Tf=
L
O"j*f
and
j=oo
are bounded on LP provided 11/p 
g(J) =
(J==
Ioj
*
N)
2
1/21 < 1/2q.
Proof. Define the operator Sj by (Sjf(~) = X6.j(6)j(~), where ~j =' (2 j +\ 2j] U [2 j , 2j +1 ).
7. The Hilbert transform along a parabola
181
Then
= where Tkf
=
=
I:
O"j
* Sj+kf.
j==
We can now argue as we did in proving (8.13) in Theorem 8.19: by applying the Plancherel theorem and (8.19) we get (8.20) Define Po by 1/2  l/po = 1/2q; equivalently, q = (po/2)'. We will first prove that (8.21) 2 ), and since the square of the lefthand side Since 100j * gl2 < II00j II (IO"j I * Igj 1 of (8.21) is the po/2 norm of E 100j * g12, by duality there exists U E Lq, Ilullq= 1, such that the lefthand side equals
L2 2;= 100j * J
gjl2U
<
02;= L2 (IO"jl * Igjl2)u < 02;= 12 IgjI20"*(u). J
J
.
Inequality (8.21) now follows by Holder's inequality and our hypothesis on 0"*. Therefore, since SiSj f = 0 unless i = j, by Theorem 8.4 and inequality (8.21),
IIT.!II"" < C
(~= laj * Sj+kfI (~= ISj+kfI
2
2 ) 1/2
)
1/2 Po
pO
< °llfllpo' If we interpolate between this and inequality (8.20) and sum in k we get that T is bounded on LP for 2
Finally, the boundedness of g(1) follows by the same argument as in the proof of Corollary 8.22. 0
8. LittlewoodPaley Theory and Multipliers
182
Theorem 8.26. Let {ILj} be a sequence of positive Borel measures on JR2 with IIILj II < C and such that for some a > 0
Ijlj(~)1 < CI2 j 61 a , Ijlj (~)  jlj (0,6) I < CI2 j 61 a .
(8.22) (8.23) Let
M2f
= s~p IlL] * fl J
be a maximal operator in the variable X2, where the measures IL] have Fourier transforms jlj(O, 6). If M2 is bounded on LP(JR), 1 < p < 00, then the maximal operator Mf(x)
= sup IILj * f(x)1 j
is also bounded on LP(JR 2), 1
Proof. Fix ¢ (jj by
E
< p < 00.
S(JR) such that ¢(O) = 1, and for each j define the measure §j(~) = jlj(~)  jlj(0,6)¢( 2j 6).
Then the (ji's satisfy the hypotheses of Theorem 8.25: for the estimate with negative exponent in (8.19) use (8.22) and bound 1¢(t)1 by Cta; for the estimate with positive exponent, add and subtract jlj(O,6) and apply the mean value theorem. Let 9 and (j* be the associated operators. Then if we argue exactly as we did in the proof of Corollary 8.23 we get that (8.24) (8.25)
< g(f)(x) + CM2 M lf(x), (j*(f)(x) < Mf(x) + CM~Mlf(x), Mf(x)
where Ml is the HardyLittlewood maximal operator in the first variable. By the Plancherel theorem, g(f) is bounded on L2. Therefore, by (8.24) and our hypothesis on M 2 , M is also bounded on £2, so by (8.25), (j*(f) is bounded on L2. Hence, by Theorem 8.25, g(f) is bounded on LP for 4/3 < p < 4. But then from (8.24) and (8.25) we have that M and (j* are also bounded on this range. Theorem 8.25 then gives us that 9 is bounded on 8/7 < p < 8. If we apply this "bootstrapping" argument repeatedly, we get that M, (j* and 9 are bounded on £P, 1 < p < 00. 0 In order to apply these theorems to the operators Hr and M r , we need to estimate the Fourier transforms of the measures O"j and ILj defined in (8.17) and (8.18). We can estimate these oscillatory integrals using Van der Corput's lemma; because this result is of independent interest we give its proof.
7. The Hilbert transform along a parabola
183
Lemma 8.27 (Van der Corput's Lemma). Let
I(a, b)
= 1b eih(t)
dt.
Then (1) if Ih'(t)1 > ).. > 0 and h' is monotonic,
II(a,b)1 <
c)..\
(2) if h E Ck([a, b]) and Ih(k) (t) > ).. > 0, 1
II(a,b)1 < C)..I/k. In either case the constants are independent of a and b. Proof. (1) If we integrate by parts we get _
I(a, b) 
l
a
b
.
I
~h (t)e
ih(t) . _dt_ _ eih(b) _ eih(a) ih'(t)  ih'(b) ih'(a)
. 1b ih(t) a e d
+~
(1) __ h'(t)
.
Therefore, since h' is monotonic,
II(a,b)1 <
l
~+
d(h'~tJ = ~+
l d(h'~t))
<
~.
(2) We will prove this for k = 2; for k > 2 the result follows by induction. First, without loss of generality we may assume that hl/(t) > ).. > O. Then h'(t) is increasing and h'(to) = 0 for at most one to E (a, b). If such a to exists, for 8 > 0 define J 1 = {t E (a , b) : t
< to  8},
h = (to  8, to + 8) n (a, b) , J3 = {t E (a, b) : t > to + 8}. (Some of these may be empty.) On Jl and h, Ih'(t)1 > ),,8, so arguing as in (1) we get
r
eih(t)dt <8()"8)1.
Jhuh The integral on J2 is less than 28, so if we let 8 result.
= ).. 1/2
we get the desired
If h' is never 0 on (a, b) then we can apply a similar argument with a or b in place of to, depending on whether h' is positive or negative. D Lemma 8.28. Let
I (b) Then for 1 < b < 2 and
=
161> 1,
1,b e
i (t6 +t',,)
II(b)1 <
dt.
CI61 1/ 2 .
8. LittlewoodPaley Theory and Multipliers
184
Proof. Let h(t) = t6 +t2 6· When 161 > 8161, Ih'(t)1 = 16 +2t61 > 161/2, so by Van der Corput's lemma, II(b)1 < C161 1 < CI61 1 / 2 , where the second inequality follows from our assumption on 6.
By the same lemma, since h"(t) = 26, II(b)1 < CI61 1/ 2 for any 6 and 6. When 161 < 8161, this implies that II(b)1 < CI61 1/ 2 . 0 We can now use Theorems 8.25 and 8.26 to prove the desired result. Corollary 8.29. The operators Hr and Mr are bounded on LP(JR2) for 1 p < 00 and 1 < p < 00 respectively.
<
Proof. Let aj, ILj be as in (8.15) and (8.16). Then by Lemma 8.28, for 161> 1 we have that
(To get the estimate for 00 use integration by parts.) Then from the relations fJ,j(~)
.
2·
= fJ,0(2 J6, 2 J6),
oj(~)
.
2·
= 00(2J6, 2 J6),
we get
The estimates
IOj (~) I, lfJ,j(~)  fJ,j (0,6) 1 < CI2 j 61 follow immediately from the definition. The maximal operator M2 of Theorem 8.26 here becomes sup j
'~l
2J
r
J2j~ltl~2Hl
g(X2  t 2) dt . ,
If we make the change of variables s = t 2 we see that M2 is dominated by the onedimensional HardyLittlewood maximal function and so is bounded on LP(JR) , 1 < p < 00. Therefore, by Theorem 8.26, Mr is bounded on LP(JR2) , 1 < p < 00.
Finally, the maximal operator a* in Theorem 8.25 satisfies a* (j) < Mr f, so by that result it follows that Hr is bounded on LP(JR 2), 1 < p < 00. 0
8. Notes and further results 8.1. References.
The original results of J. E. Littlewood and R. A. E. C. Paley are found in Theorems on Fourier series and power series, I (J. London Math. Soc. 6 (1931), 230233) and Theorems on Fourier series and power series, II (Proc. London Math. Soc. 42 (1936), 5289). They used techniques from
8. Notes and further results
185
complex variables. (See Zygmund [21].) The approach taken here is derived from A. Benedek, A. P. Calderon and R. Panzone (Convolution operators with Banachspace valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356365). Also see J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea (Calder6nZygmund theory for operatorvalued kernels, Adv. in Math. 62 (1986), 748.) The Hormander multiplier theorem first appeared in Estimates for translation invariant operators in LP spaces (Acta Math. 104 (1960),93139). The Marcinkiewicz multiplier theorem can be found in Sur les multiplicateurs des series de Fourier (Studia Math. 8 (1939), 7891). The proofs given in the text are different from the originals. In Hormander's theorem it can also be shown that the multiplier is weak (1,1). BochnerRiesz multipliers were introduced by S. Bochner (Summation of multiple Fourier series by spherical means, Trans. Amer. Math. Soc. 40 (1936), 175207). The methods in Sections 6 and 7 are adapted from J. Duoandikoetxea and J. L. Rubio de Francia .(Maximal" and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541561). For references on singular integrals and maximal functions on curves, see Section 8.8 below. Van der Corput's lemma originally appeared in Zahlentheoretische Abschiitzungen (Math. Ann. 84 (1921), 53;79). There are other versions of the LittlewoodPaley theory (with different square functions and with continuous parameter) which can be found, for example, in Stein [15, Chapter 4]. Theorem 8.6 can be thought of as a characterization of LP spaces via the LittlewoodPaley decomposition given by {Sjf}. In a similar way many other spacesincluding Besov spaces and TriebelLizorkin spacescan be characterized using different mixed norm inequalities for the LittlewoodPaley decomposition. For more information see, for instance, the books by M. Frazier, B. Jawerth and G. Weiss (LittlewoodPaley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics 79, Amer. Math. Soc., Providence, 1991) and H. Triebel (Theory of function spaces II, Birkhiiuser, Basel, 1992). 8.2. Other LittlewoodPaley inequalities. Theorem 8.30. Let {Ij } be any sequence of disjoint intervals in lR, and define the operator Sj by (Sjff (~) = Xlj (~)j(~). Then for 2 < P < 00,
(I: IS;!I') J
< Cpll!llp.
1/' P
When I j = (j,j + 1) this theorem was first proved by L. Carleson (On the LittlewoodPaley theorem, Report, MittagLeffler Inst., Djursholm,
8. LittlewoodPaley Theory and Multipliers
186
1967); new proofs were later given by A. Cordoba (Some remarks on the LittlewoodPaley theory, Rend. Circ. Mat. Palermo 1 (1981), suppl., 7580) and J. L. Rubio de Francia (Estimates for some square functions of LittlewoodPaley type, Publ. Mat. 27 (1983), 81108). The general result is due to J. L. Rubio de Francia (A LittlewoodPaley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1 (1985), 114), and another proof is due to J. Bourgain (On square functions on the trigonometric system, Bull. Soc. Math. Belg. Ser. B 37 (1985), 2026). By taking iN = X[O,N) one can see that this result is false if 1 < p < 2. A generalization of Theorem 8.30 in ~n is due to J. L. Journe (Calder6nZygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), 5591). His proof is quite difficult, especially when compared to the proof for n = 1. Simpler proofs were later given by P. Sjolin (A note on LittlewoodPaley decompositions with arbitrary intervals, J. Approx. Theory 48 (1986), 328334), F. Soria (A note on a LittlewoodPaley inequality for arbitrary intervals in JR.2, J. London Math. Soc. 36 (1987), 137142), and S. Sato (Note on a LittlewoodPaley operator in higher dimensions, J. London Math. Soc. 42 (1990),527534). Theorem 8.30 can be used to improve the Marcinkiewicz multiplier theorem (Theorem 8.13). Theorem 8.31. If m is bounded and m 2 is of bounded variation on each dyadic interval in JR., then m is a multiplier on LP, 1 < p < 00.
This result is due to R. Coifman, J. L. Rubio de Francia and S. Semmes (Multiplicateurs de Fourier de LP(IR) et estimations quadratiques, C. R. Acad. Sci. Paris 306 (1988),351354). In IR2 one can prove an "angular" LittlewoodPaley theorem: let f:1j be the region bounded by the lines with slopes 2j and 2j + 1 , and define Sj by (Sjfr(~) = X~j (~)i(~). Then for 1 < p < 00,
(I: ISjfl t 2
J
2
< Cpllfllp· P
This result is due to A. Nagel, E. M. Stein and S. Wainger (Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 10601062). There are also weighted versions of the LittlewoodPaley inequalities. Theorem 8.6 is true in LP(w) when w E Ap. This was proved by D. Kurtz (LittlewoodPaley and multiplier theorems on weighted LP spaces, Trans. Amer. Math. Soc. 259 (1980), 235254) for 7/J supported in an annulus, but his result can be extended to 7/J as in Theorem 8.6. The proof of Theorem 8.30 by Rubio de Francia actually gives a weighted version with A p / 2 weights when p > 2; the proof is based on techniques introduced in his paper
8. Notes and further results
187
with Ruiz and Torrea cited in Section 8.1. The same result in higher dimensions was proved by H. Lin (Some weighted inequalities on product domains, Trans. Amer. Math. Soc. 318 (1990), 6985). It is not known whether the weighted version of Theorem 8.30 is true in L2(w) with wEAl; however, it is true in the special case of equally spaced intervals (see the paper by Rubio de Francia in Publ. Mat. cited above.)
8.3. The multiplier of the ball and BochnerRiesz multipliers.
In dimension n behavior.
> 2, the multiplier of the ball exhibits the worst possible
Theorem 8.32. The characteristic function of a (Euclidean) ball in n > 2, is not a multiplier on LP if p # 2.
~n,
This negative result was proved by C. Fefferman (The multiplier problem for the ball, Ann. of Math. 94 (1972), 330336). This is much stronger than what can be proved using Lemmas 8.17 and 8.18: this approach only shows that it is not a multiplier if !l/p  1/2! > 1/2n. However, it is a multiplier when restricted to radial functions in LP if !l/p  1/2! < 1/2n. This is due to C. S. Herz (On the mean inversion of Fourier and Hankel transforms, Proc. Na~. Acad. Sci. U.S.A. 40 (1954), 996999). For a > 0, the sufficient condition in Theorem 8.14 was first proved by E. M. Stein (Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482492) using Theorem 1.22. In this result we omit a range of values for p when 0 < a < (n  1)/2. Despite the negative result for a = 0 (the multiplier of the ball), it is conjectured that T a is a bounded operator on LP when
(8.26)
1
P
1
2" <
2a + 1
2n
This has only been fully proved in ~2. The first proof was given by L. Carleson and P. Sjolin (Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287299). Additional proofs were given by L. Hormander (Oscillatory integrals and multipliers on F LP, Ark. Mat. 11 (1973), 111), C. Fefferman (A note on spherical summation multipliers, Israel J. Math. 15 (1973), 4452) and A. Cordoba (A note on BochnerRiesz operators, Duke Math. J. 46 (1979), 505511). In higher dimensions, the first result proving the boundedness of T a for the full range of values ofp given by (8.26) for some values of a < (n1)/2 is due to C. Fefferman (Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 936). He showed it for a> (n1)/4. An important
8. LittlewoodPaley Theory and Multipliers
188
aspect of his proof is that it introduced the connection between BochnerRiesz multipliers and the restriction of the Fourier transform to spheres. In the next section we give stronger results derived using this connection. All of these results can be found in the book by K. M. Davis and Y. C. Chang [3]. BochnerRiesz multipliers give rise to a family of summability methods for the Fourier transform. Besides norm convergence, we can also consider pointwise convergence, that is, whether lim TRf(x)
(8.27)
R+=
= f(x)
a.e.,
where
The usual way to study pointwise convergence is via the associated maximal operator (Ta)* f(x) = sUPR>O ITRf(x)1 (see Chapter 2, Section 2). For a greater than the critical index (n  1)/2 , (Ta)* is bounded by the HardyLittlewood maximal operator and pointwise convergence follows immediately. For a < (n  1)/2 partial results are known. When n = 2 and p > 2, (8.27) holds in the range given by (8.26). See the paper by A. Carbery (The boundedness of the maximal BochnerRiesz operator on L4(JR 2), Duke Math. J. 50 (1983), 409416). The same result for n > 3, p > 2, and a > (n  1) /2( n + 1) is due to M. Christ (On almost everywhere convergence of BochnerRiesz means in higher dimensions, Proc. Amer. Math. Soc. 95 . (1985), 162'0). Both results were proved by showing that (Ta)* is bounded on LP for the corresponding values of p. A. C arb ery, J. L. Rubio de Francia and L. Vega (Almost everywhere summability of Fourier integrals, J. London Math. Soc. 38 (1988), 513524) proved that (Ta)* is bounded on the weighted space L 2(lxl,B) if 0 < f3 < 1 + 2a < n, and deduced almost everywhere convergence in these spaces. For those p > 2 such that (8.26) holds, there exist f3 < 1 + 2a for which £P c L2 + L2(lxl,B), so (8.27) holds for all such p. Note that this gives us the almost everywhere convergence of BochnerRiesz means on a larger range of values of p than that for which we know the boundedness of T a (and a fortiori of (Ta)*).
8.4. Restriction theorems.
i
is continuous and bounded, so for any S c ]Rn, Ilill£<)O(5) < IlfIILl(JRn). In general, however, if f E LP(JRn ) for some p > 1 and S has measure 0 we cannot, a priori, define on S since it is only defined almost everywhere. But if for, say, f E C~ we have the estimate IliIILq(5) < Cp~qllfIILP(JRn), then by continuity, if f E LP we can define the restriction of f to S as an Lq function. If S is contained in a hyperplane If fELl then
i
8. Notes and further results
189
then this estimate only holds for p = 1. Surprisingly, it is possible to get nontrivial restriction estimates for some S with nonzero curvature. The case S = sn1, the unit sphere in IR n , is of particular interest because it is related to BochnerRiesz multipliers as the following theorem of C. Fefferman shows (see the Acta paper cited above). Theorem 8.33. If for some p
> 1,
(8.28) then.ya is bounded on LP for all a such that (8.26) holds, C. Fefferman proved the restriction inequality for 1 < p < 4n/(3n + 1), which gives the condition a > (n  1) /4 mentioned above. This result was improved by P. Tomas (A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477478), developing earlier work of E. M. Stein. Theorem 8.34 (TomasStein). Inequality (8.28) holds for all p such that
2n+2 l

n+3
As an immediate consequence we have the boundedness of BochnerRiesz multipliers in the full range (8.26) if a> (n  1)/(2n + 2). It is conjectured that, in general,
(8.29) is true if and only if
2n l
and
I
n+l nl
p>q.
That p and q must be restricted to this range can be seen by taking f to be the characteristic function of a rectangle "adapted" to the sphere, that is, a rectangle with sides 0 x 01 / 2 x· .. X 01 / 2 centered at (1,0, ... ,0). When n = 2 this conjecture was proved by C. Fefferman in the Acta paper cited above. Also see the paper by A. Zygmund (On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189201). It is also true when q = 2; this is shown by Theorem 8.34. Theorem 8.34 was the best known result for n > 3 until it was improved by J. Bourgain (Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), 147187). He showed that the restriction conjecture is true for 1 < p < p( n), where p( n) is defined inductively and (2n + 2)/(n + 3) < p(n) < 2n/(n + 1)). (For instance, p(3) = 31/23.)
190
8. LittlewoodPaley Theory and Multipliers
There is a close connection between the restriction problem, BochnerRiesz multipliers, and the Kakeya maximal operator (seeChapter 2, Section 8.9). All the previously known results for these problems were improved in Bourgain's paper. Further improvements for the BochnerRiesz conjecture and the restriction problem are again due to J. Bourgain (Some new estimates on oscillatory integrals, Essays on Fourier Analysis in Honor of Elias M. Stein, C. Fefferman, R. Fefferman and S. Wainger, eds., pp. 83112, Princeton Univ. Press, Princeton, 1995). In the case n = 3, these were in turn improved by T. Tao, A. Vargas, and L. Vega (A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 9671000). Very recently, T. Tao proved a kind of converse of Theorem 8.32 (The BochnerRiesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), 363376). 8.5. Weighted norm inequalities for multipliers.
Given a multiplier m and the associated operator T m , we can try to characterize, just as we did in Chapter 7 for singular integrals, the weights w such that Tm is a bounded operator on LP(w). The Marcinkiewicz multiplier theorem has a straightforward generalization to the weighted case. Theorem 8.35. Let m be a bounded function which has uniformly bounded variation on each dyadic interval in ~. Then if w E A p , T m is a bounded operator on: LP (~, w), 1 < p < 00.
This result was proved by D. Kurtz; see the paper cited in Section 8.2 above. In it he also proved a weighted analogue of Theorem 8.14. For the multipliers in Hormander's theorem' (Corollary 8.11) the dition is no longer sufficient; however, the following is true.
Ap
con
Theorem 8.36. For n > 2, let k = [n/2] + 1. Suppose n/k < p < 00 and w E A pk / n , or 1 < p < (n/k)' and wIpI E Aplk/n Then, ifm is a multiplier which satisfies (8.10), Tm is a bounded operator on LP(w). lfn > 2 then we may extend the range ofp's to include the endpointsp = n/k andp = (n/k)'.
Theorem 8.36 was proved by D. Kurtz and R. Wheeden (Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343362). Much less is known about weighted norm inequalities for BochnerRiesz multipliers. For a > (n  1)/2, it is known that T a is bounded on LP(w), 1 < p < 00, when w E Ap. When a > (n  1)/2 it can be shown that ITa f(x)1 < eM f(x), where M is the HardyLittlewood maximal operator,
8. Notes and further results
191
so this follows at once from Theorem 7.3. For the critical index a = (n1)/2, this is due to X. Shi and Q. Sun (Weighted norm inequalities for BochnerRiesz operators and singular integral operators, Proc. Amer. Math. Soc. 116 (1992), 665673). A different proof follows from results in the paper by Duoandikoetxea and Rubio de Francia cited above. In this case the operator is also weak (1,1) with respect to Al weights. See the paper by A. Vargas (Weighted weak type (1,1) bounds for rough operators, J. London Math. Soc. 54 (1996), 297310). For some partial results in the general case, see the paper by K. Andersen (Weighted norm inequalities for BochnerRiesz spherical summation multipliers, Proc. Amer. Math. Soc. 103 (1988), 165171). 8.6. The spherical maximal function.
In Corollary 8.24 we proved that the dyadic spherical maximal function is bounded on LP, 1 < p < 00, for n > 2. This was first proved independently by C. Calderon (Lacunary spherical means, Illinois J. Math. 23 (1979),476484) and R. Coifman and G. Weiss (Review of the book LittlewoodPaley and multiplier theory by R. E. Edwards and G. I. Gaudry, Bull. Amer. Math. Soc. 84 (1978), 242250). When we consider the maximal function with continuous parameter,
Mf(x) = sup (
f(x  ty) d(j(y)
t>O } Snl
the following result is true. Theorem 8.37. M is bounded on
LP(n~n)
if and only if p > n/(n  1).
For n > 3, Theorem 8.37 was proved by E. M. Stein (Maximal functions: Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 21742175), and for n = 2 by J. Bourgain (Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 6985). Other proofs for n > 3 are due to M. Cowling and G. Mauceri (On maximal functions, Rend. Sem. Mat. Fis. Milano 49 (1979), 7987) and J. L. Rubio de Francia (Maximal functions and Fourier transforms, Duke Math. J. 53 (1986),395404). This last proof is close in spirit to the method developed in Section 6: a decomposition of the multiplier into dyadic pieces, an L2 estimate for each piece with exponential decay, and an LI estimate derived with vectorvalued techniques. In the case n = 2, new proofs were given by G. Mockenhaupt, A. Seeger and C. Sogge (Wave front sets, local smoothing and Bourgain's circular maximal theorem, Ann. of Math. 136 (1992), 207218) and W. Schlag (A geometric proof of the circular maximal theorem, Duke Math. J. 93 (1998), 505533).
8. LittlewoodPaley Theory and Multipliers
192
Weighted norm inequalities for the spherical maximal operator (with both discrete and continuous parameters) are in J. Duoandikoetxea and L. Vega (Spherical means and weighted inequalities, J. London Math. Soc. 53 (1996),343353). 8.7. Further results for singular integrals.
The arguments in Section 7 readily extend to prove additional results about singular integrals which cannot be proved using the method of rotations. For example, define the operator T to be convolution with the kernel
O(x')
K(x) = h(lxl)W' where 0 E Lq(snI) for some q the growth condition sup Rl R>O
> 1 and has zero integral, and h satisfies
ior
R
Ih(t)1 2 dt <
00.
Then the proof of Theorem 8.19 can be readily adapted to show that T is bounded on LP, 1 < p < 00. Singular integrals of this type were first considered by R. Fefferman (A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), 266270), and J. Namazi (A singular integral, Proc. Amer. Math. Soc. 96 (1986),421.,424). Similar techniques can also be used to prove weighted norm inequalities. Theorem 8.38. Given 0 E L=(snI) such that JO(u) da(u) the singular integral
Tf(x) If W E A p, 1
r
= p. V. iJRn
O(y') lYF f(x 
=
0, define
y) dy.
< p < 00, then T is bounded on LP (w ) .
The proof of this result can be found in the paper by Duoandikoetxea and Rubio de Francia cited above. The hypothesis that 0 E L= cannot be weakened to 0 E Lq, q < 00; counterexamples were first given by B. Muckenhoupt and R. Wheeden (Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249258). When o E Lq, 1 < q < 00, T is bounded on LP(w) if q' < p < 00 and w E Apjql or if 1 < p < q and wIpi E Apljql. (Cf. Theorem 8.36 above.) This condition was discovered independently by J. Duoandikoetxea (Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869880) and D. Watson (Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), 389399). Additional weighted inequalities for these operators, together with structural properties of the weights, are given by D. Watson (Vectorvalued
8. Notes and further results
193
inequalities, factorization, and extrapolation for a family of rough operators, J. Funct. AnaL 121 (1994), 389415) and J. Duoandikoetxea (Almostorthogonality and weighted inequalities, Contemp. Math. 189 (1995), 213226). Weighted weak (1,1) inequalities were proved by Vargas in the paper cited in Section 8.5. ·8.8. Operators on curves.
Besides the motivating example described in Section 7, singular integrals and maximal functions on curves have appeared in several different contexts and have been studied extensively. The first results were for homogeneous curves (curves of the type (tal, t a2 , ... , tan) and some more general versions); these results were later extended to "wellcurved" curves: curves r whose derivatives r'(O), r"(O), ... span JR n . The survey article by E. M. Stein and S. Wainger (Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 12391295) contains the principal results and their proofs. Another good summary is in the article by Wainger (Averages and singular integrals over lower dimensional sets) in [16]. The next case considered was that of "fiat" curvesthat is, curves whose derivatives at the origin all vanish. These were first considered in ]R2; see (among others) the article by Duoandikoetxea and Rubio de Francia cited above, the articles by M. Christ (Hilbert transforms along curves, II: A fiat case, Duke Math. J. 52 (1985), 887894), A. Nagel, J. Vance, S. Wainger and D. Weinberg (Hilbert transforms for convex curves, Duke Math. J. 50 (1983), 735744, and Maximal function for convex curves, Duke Math. J. 52 (1985), 715722), H. Carlsson et al. (LP estimates for maximal functions and Hilbert transforms along fiat convex curves in ]R2, Bull. Amer. Math. Soc. 14 (1986), 263267), A. Cordoba and J. L. Rubio de Francia (Estimates for Wainger's singular integrals along curves, Rev. Mat. Iberoamericana 2 (1986), 105117), and A. Carbery, M. Christ, J. Vance, S. Wainger, and D. Watson ( Operators associated to fiat plane curves: LP estimates via dilation methods, Duke Math. J. 59 (1989), 675700). The extension of the results on fiat curves to higher dimensions has been more difficult. See the articles by A. Nagel, J. Vance, S. Wainger and D. Weinberg (The Hilbert transform for convex curves in JRn , Amer. J. Math. 108 (1986),485504), A. Carbery, J. Vance, S. Wainger and D. Watson (The Hilbert transform and maximal function along fiat curves, dilations, and differential equations, Amer. J. Math. 116 (1994), 12031239), A. Carbery, J. Vance, S. Wainger, D. Watson and J. Wright (LP estimates for operators associated to fiat curves without the Fourier transform, Pacific J. Math. 167 (1995), 243262), and A. Carbery and S. Ziesler (Hilbert transforms and maximal functions along rough fiat curves, Rev. Mat. Iberoamericana 10 (1994), 379393). Also see the survey article by A. Carbery, J. Vance,
194
8. LittlewoodPaley Theory and Multipliers
S. Wainger and D. Watson (Dilations associated to fiat curves in ~n, Essays on Fourier Analysis in Honor of Elias M. Stein, C. Fefferman, R. Fefferman and S. Wainger, eds., pp. 113126, Princeton Univ. Press, Princeton, 1995). In recent years nonconvolution operators have also been studied. In this case the operators are defined on variable curves, that is, curves which also depend on the point x. In a similar fashion, curves have been replaced by surfaces, both in the convolution and the variable case. Some examples can be found in Stein [17, Chapter 11].
Chapter 9
The Tl Theorem
1. Cotlar's lemma In this chapter we return to a question fwe first considered in Chapter 5: given a Calder6nZygmund operator T with associated kernel K, when i~ T bounded on L2? If T is a convolution operator, the problem reduces to showing that K E L=. For operators which are not convolution operators the problem is much more difficult. In some cases, however, a result due to M. Cotlar is useful. Lemma 9.1 (Cotlar's Lemma). Let H be a Hilbert space, {Tj } a sequence of bounded linear operators on H with adjoints {TJ}, and {a(j)} a sequence of nonnegative numbers such that
'hen for all integers nand m, n
< m,
=
m
Proof. Let m

195
196
9. The T1 Theorem
Then But
11811 = 1188*11 1/2; in fact,
for any integer k
> 0, IISII = II(SS*)kI11/2k.
m
(88*)k
=
L
jl, ... ,j2k=n
Tjl T;2 .. , Tj2kl Tj*2k'
and the norm of each summand can be bounded in two ways:
II Til Tj*2 ... T12k  l Tj*2k II < IITjl T;211· . ·IIT12k  l T;2k II < a(j1  j2) ... a(j2k1  hk), and
IITjl Tj*2 ... T12k  l Tj*2k II < IITjlIIIITj*2Tj311"
·IIT;2k_2T12klIIIITj*2k II
< a(0)1/2a(h  j3) ... a(j2k2  j2k_1)a(0)1/2. If we take the geometric mean of both bounds and sum over all the ji we get m
II(SS*)kll < a(0)1/2
L
a(h  j2)1/2a(j2 
j3)1/2 ... a(j2k1 
j2k)1/2.
jl, ... ,12k=n Fix
j1,'"
j1'
Then
,hk1 and sum over j2k; then sum over j2k1 and so on down to
D(SS·)kll < a(O)I/2(m  n + 1) where m  n Hence,
(
i%;=
a(i)I/2 )
2k1 ,
+ 1 is the number of terms which ,appear when we sum in j1. 00
IISII
=
II (SS*)kI11/2k <
(m  n
+ 1)1/2k
L
a(i)1/2,
i=oo
and if we let k tend to infinity we get the desired result.
o
If for all x in a dense subset of H the sum
00
exists, we can extend it to an operator on all of H; the norm of this operator is bounded by Ea(i)1/2. In fact, it is not necessary to assume that ETjx exists on a dense subset because the hypotheses of Cotlar's lemma imply that the series converges for any x E H.
2. Carleson measures
197
Example 9.2 (An application to the Hilbert transform). Let H = L2(IR) and for f E L2 (IR) define
Tjf(x)
=
r
12
f(x  t) dt.
t
j
For each integer j, ITjf(x)1 < 4Mf(x) (where M is the HardyLittlewood maximal operator), so T j is clearly bounded on L 2 (IR). We will use Cotlar's lemma to prove that any finite sum of the Tj's is uniformly bounded, which in turn proves that the truncated integrals of the Hilbert transform are uniformly bounded. (Actually, we will show that only those integrals truncated at the dyadic numbers 2j are bounded, but it is easy to pass to any other truncation. ) Since Tj = Tj, we only need to estimate IITiTjll. If we let Kj(x) = 1 X X.6.j(x), where ~j = {x E IR: 2j < Ixl < 2j +1}, then Tjf(x) = K j * f(x) and TiTjf(x) = Ki * K j * f(x). Hence,
IITiTjl1 < IIKi * Kj 11 1 . However,
Ki
r1
* Kj(x) = JJR i X.6.
1
i
(t) x _ tX.6.j (x  t) dt.
Without loss of generality we may assume that i < j and x > O. Then Ki * Kj(x) = 0 if x fj. (2j  2i+1, 2j + 1 + 2i+1), and on this interval, IKi * K j (x) I < 2 . 2 j . We will use this estimate on the intervals (2j  2i+ 1, 2 j + 2i+1) and (2 j +1 _ 2i+ 1, 2j + 1 + 2i+ 1). On the rest of the interval, if t E
Ki * Kj(x) hence, IKi
=
JII:" ~X",(t)
~i
then x  t E
C
1t 
~j,
so that
~) X,,; (x 
t) dt;
* Kj(x)1 < 2· 2i 2j . Therefore, IIKi * Kjl11 < C2 li jl ,
and this estimate allows us to apply Cotlar's lemma as desired.
2. Carleson measures Definition 9.3. A positive measure v on IR~+l is a Carleson measure if for every cube Q in IRn
v(Q
x (0, l(Q))) < CIQI,
where l(Q) is the side length of Q. The infimum of the possible values of the constant C is called the Carleson constant of v and is denoted by II v II.
9. The T1 Theorem
198
An example of a Carleson measure in the upper halfplane IR~ is drdB (polar coordinates). The property underlying the definition of Carleson measures is not limited to cubes but also extends to more general sets. Given an open set E c IR n , let
E = {(x, t)
E IR~+l :
B(x, t) C E}.
Then the following is true. Lemma 9.4. If v is a Carleson measure in IR~+l and E C IR n is open, then
v(E) <
CllvlllEI.
Proof. In IR n , form the Calder6nZygmund decomposition of the characteristic function of E at height 1/2. This yields a collection of disjoint dyadic cubes {Qj} such that
E
C
UQj
and
lEI < L IQjl < 21EI·
j
j
Let (x, t) E E; then x E Qj for some j. If Qj is the dyadic cube containing Qj whose sides are twice as long, then Qj contains points of E e . Therefore, since B(x, t) C E, we must have that t < .Jril(Qj) = 2y'nl(Qj). Hence,
E C UQj x (0,2ynl(Qj)), j
and so j
j
o The converse of Lemma 9.4 is also true. It is straightforward to show that if for any ball B C IRn, v(B) < OIBI, then v satisfies Definition 9.3, and Ilvll is comparable with the constant in this inequality. A Carleson measure can be characterized as a measure v for which the Poisson integral defines a bounded operator from LP(IRn, dx) to LP(IR~+l, v). This is a consequence of a more general result. Theorem 9.5. Let ¢ be a bounded, integrable function which is positive, radial and decreasing. For t > 0, let ¢t(x) = tn¢(t1x). Then a measure v is a Carleson measure if and only if for every p, 1 < p < 00,
(9.1)
(I¢t * f(x)IP dv(x, t) < 0
jIT?,.n+l
+
The constant 0 is comparable with
Ilvll.
r lJ(x)I Pdx.
jIT?,.n
2. Carleson measures
199
Proof. First suppose that v is a Carleson measure. Define the maximal operator
M¢f(x) = sup{l¢t * f(y)1 : Ix  yl < t}. It is easy to show (cf. Chapter 2, Section 8.7) that
M¢f(x) < CM f(x), whereM is the HardyLittlewood maximal operator. Now (9.2)
{ I¢t J~n+l
* f (x ) IP dv (x, t)
+
=p Let E>.
= {x
E ~n :
10= AP1V( {(x, t)
E
~~+l
: I¢t
* f(x)1 > A}) dA.
M¢f(x) > A}; then
{(x, t)
~~+l : I¢t
E
* f(x)1 > A} C E\.
Hence, by Lemma 9.4
v({(x,t)
E
~~+l: l¢t*f(x)1 > A}) < v(E>.) < CllvIIIE>.I.
Therefore, the righthand side of (9.2) is dominated by _ pCllvl1 {= AP1IE>.1 dA
Jo
=
Cllvll { M¢f(x)P dx < Cllvll { If(x)IP dx,
J~n
J~n
which gives us inequality (9.1). Now, conversely, suppose that (9.1) holds. Let B C ~n be the ball with center Xo and radius r, and let (x, t) E B (Le. B(x, t) C B). Then I¢t
* XB(x)1 = (
J B(xo,r)
¢t(x  y) dy > ( ¢t(Y) dy = ( . ¢(y) dy = A. J B(O,t) J B(O,l)
Hence,
v(B) < Al ( I¢t * XB(x)I PdV(x, t) < C { IXB(X)I Pdx = CIBI, J~n P J~n+l +
so by the remark following Lemma 9.4, v is a Carleson measure.
0
Carleson measures can also be characterized in terms of BMO functions. Theorem 9.6. Let b E BMO and let 'lj; E S be such that
the measure v defined by dv = Ib * 'lj;t (x ) 12 dx dt
t
is a Carleson measure such that Ilvll is dominated by Ilbll;.
J'I/J =
O. Then
9. The T1 Theorem
200
Proof. Fix a cube Q in ffin. Since the translate of a BMO function is again in BMO, we may assume without loss of generality thatQ is centered at the origin. Let Q* be the cube with the same center whose side length is 2fo times larger. Let bQ* be the average of b on Q*. Then, since J'ljJ = 0,
= 11 * ° 11° + 11 ° l (Q)
v(Q x (0, Z(Q»)
dxdt
Ib ?/;t(x) 12 
t
Q
l (Q)
<2
dxdt
I(b  bQ* )XQ*
* ?/;t(x) 12 
Q
l (Q)
2
t dxdt
I(b  bQ* )X{lRn\Q*}
* ?/;t(x) 12 
Q
t
=h+h.
(Recall that Z(Q) denotes the side length of Q.) Now,
h < 2 r roo I((b _ bQ* )XQ* r(~)121~(t~)12 d~ dt; ~h t since ~(O) = 0, 1~(t~)1 < Cmin(lt~l, It~ll), and so
1°00
dt 11/1~1 dt /,00 It~I2 dt 1?/;(t~)12  < C. It~12  + C tot l/I~I t Therefore, by this estimate and by Corollary 6.12, A
< C.
h < Crib  bQ* 12 < CIQlllbll;.
JQ*
To estimate h, let Q k denote the cube centered at the origin whose side length is 2k times that of Q*. Then
h <
11
l (Q)
L00 1
2
(b(y)  bQ* )?/;t(x  y) dy
dxdt t
Q ° k = O Q k+1 \Q k
Since x E Q, if y (j. Qk then Ix  yl > 2k  1 Z(Q). Therefore, because?/; E S, I?/;t(x  y) I < Ct n (t 1 2k Z( Q) )nl.
Hence,
r
I2
< C Jc
Q
rl(Q)
In
°
k10
l(Q)
00
r
L Jc
k=O (
00
t Ib(y)  bQ* I dy (2 k Z(Q»n+l
*
Q k +1
t
~ 2k l(Q) Ilbll,
2 )
2
dxdt
t
dx dt
t
< CIQlllbll;·
o
3. Statement and applications of the T1 theorem
201
The converse of Theorem 9.6 is also true, although we will not prove it here: if'ljJ and v are as above and v is a Carleson measure then b E BMO. For a proof see Stein [17, Chapter 4]. Finally, if we combine Theorems 9.5 and 9.6 we get the following. Corollary 9.7. Let
{ ioroo I
(T j, g)
= {
( K(x, y)j(y)g(x) dx dy. i]Rn i]Rn
The adjoint operator T* is defined by
(T* j, g)
= (I, Tg),
j, gE S(JR n ).
It is associated with the standard kernel K(x, y)
= K(y, x).
When T is a bounded operator on L2 we can define T j for j E Loo as a function in BMO (see Chapter 6, Section 2). Now, however, if we want to define T on L oo , we cannot use this approach since we do not know a priori that T is bounded on L2. As a substitute we will define T j for functions j which are bounded and in Goo. In fact, T j will be an element of the dual of G:;O, , the subspace of C~ functions with zero average.
OO Fix 9 E C:;O , with support in B(O, R) and let j E L n Coo. Fix 'l/JI, 'l/J2 E COO(JR n ) such that supp('l/JI) c B(O, 3R), 'l/Jl = 1 on B(O, 2R) and such that 'l/Jl + 'l/J2 = 1. Then j'I/Jl E S(JR n ) and (T(j'I/Jl) , g) makes sense. If j were a function of compact support then by (9.3) we would have that
(T(j'I/J2) , g)
=
{ { I((x, y )j(y )'l/J2 (y )g(x) dx dy. i]Rn i]Rn
9. The T1 Theorem
202
Since j need not have compact support, we instead define (T(j'l/J2) , g) by
r r [K(x, y)  K(O, y)]j(y)'l/J2(y)g(X) d~ dy.
JIRn JIRn
Since g has zero average, this coincides with the previous definition when j has compact support; for arbitrary bounded, C= functions j this expression makes sense because the integral in y is taken far from the origin (on SUpp('l/J2)), and (since K is a standard kernel)
Clxl o IK(x, y)  K(O, y)1 < Iyln+o' We now define Tj by
(9.4)
(Tj,g) = (T(j'l/JI),g)
+ (T(j'l/J2),g).
Since g has zero average, this definition is independent of the choice of 'l/JI and'l/J2. Below we will use (9.4) to define T1 and T*1. Given j E C= n L=, we say that T j E BMO if there exists a function bE BMO such that
(Tj,g) = (b,g) for all g E C~. Since C~ is dense in HI, by the duality of HI and BMO (Theorem 6.15) this is equivalent to saying that
I(T j, g) I < CIIgIIHl. To state the T1 theorem we need one more definition. Definition 9.8. The operator T has the weak boundedness property (customarily denoted W BP) if for every bounded subset B of C~(IRn) there exists a constant CB such that for any (PI, (h E B, x ERn and R > 0,
I(Trpf,R, rp~,R)1 < CBRn, where
(y  x)
rpix,R (y) = rpi I I '
i
=
1,2.
If an operator T is bounded on LP for some p then it is easy to show that it satisfies the weak boundedness property.
Given a standard kernel K which is antisymmetric (i.e. K(x, y) K(y, x)), we can associate it with an operator T by defining
(9.5)
(T j, g) = lim E+O
1
Ixyl>E
K(x, y)j(y)g(x) dy dx.
3. Statement and applications of the Tl theorem
203
By the antisymmetry of K this is equivalent to
(Tf, g)
1 = 2
r r
JITRn JITRn
K(x,y)[j(y)g(x)  f(x)g(y)]dydx,
and this integral converges absolutely: by the mean value theorem (adding and subtracting f(y)g(y)) the term in square brackets yields a factor of Iy  xl which lets us integrate near the diagonal. The operator defined by (9.5) has the W BP. In fact, it is enough to note that
(TCPf,R, ¢~,R) = R n (Tz ,R¢l, ¢2), where Tz,R is defined as in (9.5) but with the kernel R n K(Rx + z, Ry + z) . .This is a standard kernel with the same constants as the kernel K. The importance of the W BP is shown by the onedimensional operator of differentiation. It takes S(JR) to S' (lR) and is associated with the standard kernel zero, but is not bounded on £2. It is straightforward to check that it does not have the weak boundedness property. Clearly, differentiation is not the operator associated by (9.5) with the kernel zerothat would be the zero operator which trivially has the W BP. We can now state the T1 theorem. Theorem 9.9. An operator T : S(JR n ) + S'(JRn ), associated with a standard kernel K, extends to a bounded operator on L2(JRn ) if and only if the following conditions are true:
(1) T1 E BMO, (2) T*l E BMO, (3) T has the WBP. We have already seen the necessity of these three conditions: (1) and (2) follow from Theorem 6.6 and the remarks after it, and we noted the necessity of (3) immediately after Definition 9.8. Corollary 9.10. If K is a standard kernel which is antisymmetric and T is the operator defined by (9.5), then T is bounded on L2(JRn ) if and only if
T1
E
BMO.
This follows immediately from Theorem 9.9 since T always has the W BP and T*l
= T1.
Example 9.11 (An application of the T1 theorem). Recall that in Chapter 5, Section 3, we defined the Calderon commutators,
Tkf(x)
=
lim E+O
r
J1xyl>E
(A(X)  A(y))k f(y) dy, x  Y
x  Y
9. The T1 Theorem
204
where A : IR t IR is Lipschitz and k > 0 is an integer. These are operators defined as in (9.5) with antisymmetric kernels
K ( ) _ (A(x)  A(y))k k x, Y (x _ y) k+ 1 Corollary 9.12. The operators Tk are bounded on L2 and there exists a positive constant C such that IITkl1 < CkIIA'II~. It follows from this result and Definition 5.11 that the Tk's are CalderonZygmund operators.
Proof. From the proof of Theorem 9.9 we will see that the norm of an operator T depends linearly on the constants involved in the hypotheses; in the special case of Corollary 9.10 these reduce to the constants of K as a standard kernel and the B M 0 norm of Tk 1.
Since a straightforward argument shows that the constants of Kk as a standard kernel are bounded by C 1 (k + 1) IIA'II~, it will suffice to show that for some C > 0 and all k, (9.6) For in this case, if C
> C 1 then we have
We will prove (9.6) by induction. When k = 0, To equals the Hilbert transform, and so To1 = O. Now suppose that (9.6) is true for some k. If we integrate by parts we see that
(Formally, this calculation is immediate; however, making it rigorous requires some care. The details are left to the reader.) Therefore,
From the proof of Theorem 6.6 we have that
+ l)IIA'II~ + IITkIIL2,p) < C 3 (C 1 (k + 1) + 2C2Ck+l)IIA'II~.
IITkIILoo,BMO < C 3 (C 1 (k
Since Cl, C2, C3 are all independent of k, if C is sufficiently large then
This yields the desired inequality.
o
4. Proof of the T1 theorem
205
Another operator we considered in Chapter 5, Section 3, is associated with the antisymmetric kernel
K(x, y)
=
1 x  y + i(A(x)  A(y))'
and is related to the Cauchy integral along a Lipschitz curve. As we showed there, if IIA'lloo < 1 then we can expand this kernel as 00
K(x, y)
=
I: i kKk(x, y). k=Q
From this expansion and Corollary 9.12 we get the following. Corollary 9.13. There exists E > 0 such that ij IIA'lloo < E then the operator associated with the kernel K is a Calder6nZygmund operator. For more on these operators see Section 5.2 below.
4. Proof of the Tl theorem As we already noted, all that we have to prove is that the three conditions are sufficient for T to be bounded on L2. We will consider two cases: first the case when T1 = T*l = 0, and then the general case which we will reduce to the first. Since the proof is rather long, we will try to clarify it by giving several intermediate results as lemmas.
Case 1: T1 = T*l = 0 and T has the W BP. Fix a radial function cpE S(JRn ) which is supported in B(O, 1) and has integral 1, and let cPj(x) = 2 jn ¢(2 j x). Define the operators Sjj
= cPj * j
and
f1j
= Sj
 Sjl.
Then we can formally decompose T as 00
I: (SjTf1j + f1jTSj 
T =
f1 j Tf1 j ).
j=oo
The following lemma makes this identity precise. Lemma 9.14. Let N
RN
=
I: (SjTf1j + f1jTSj 
f1j Tf1 j ).
j=N
Thenjor j,g E Cgo(JRn ), lim (RNf, g) = (Tj, g). N7OO
206
9. The Tl Theorem
Proof. If we expand the sum for RN we get RN
= SNTSN 
SN+ITSN+1 .
Since SN i converges to i in S(ffi.n), and since T is a continuous map from S(ffi.n) to S'(lRn ), lim (S_NTSN i, g) = (Ti, g).
N+oo
Therefore, we need to show that (9.7) We will prove this using the W BP for T. For N
> 1 define the functions
iN(X) = 2nN (¢ * i(2 N ·))(x).
Then {iN} is a bounded subset of Cg<'(ffi.n) since the iN'S are supported in a fixed, compact subset of lRn and satisfy the uniform bound IID a iN 1100
< lIilllIlDa¢lIoo.
Define the sequence {gN} in the same way. Then by the W BP, I(TiN(2 N .),gN(2 N .))1
< C2nN.
Since iN(2 N x) = 2nN SN i(x) and 9 satisfies the same identity, it follows that
I(TSNi, SNg) I < C2 nN .
o
If we let N tend to infinity we get (9.7).
By Lemma 9.14, to prove that T is bounded on L2 it will suffice to prove that the RN'S are uniformly bounded on L2. We will show this by applying Cotlar's lemma. For simplicity, we will only consider the operators Tj = SjT.!':lj and show that I TjT'; II, IITITk I <' C28Ijkl, where c5 comes from the standard estimates for K, the kernel of T. Similar estimates for the other two summands are gotten by analogous arguments. (Note that below we only use the hypothesis T*l = 0; in the arguments for these other operators we use the fact that T1 = 0.) We begin by finding the kernel of Tj
j(u)
=
Tin (u 2i X)
let 'l/;
:
= ¢  ¢l
and define
and 1f;J(u) = Tin'ljJ (u 2i
y) .
Then for i, 9 E Cg<'(ffi.n ), (Tji,g)
=
(TtJ.ji,Sjg) =
rr
JlR JlR n
(T'l/;],¢j)i(y)g(x)dydx.
n
Hence, the kernel K j of Tj is K j (x, y) = (T'l/;] , ¢j).
4. Proof of the
T1 theorem
207
= (1+lxl)n8, where b is the constant from the estimates for K as a standard kernel, and let pj(x) = 2 jn p(2 j x). Then
Lemma 9.15. Definep(x)
(1) IKj(x, y)1 < CPj(x  y); (2) IKj(x, y)  Kj(w, y)1 < Cmin(1, 2 j lx  wl)[Pj(x  y) and the analogous inequality in the second variable; (3) for all x,
J.
Kj(x, y) dy = 0;
r
Kj(x, y) dx = O.
Rn
(4) for all y,
+ pj(w 
y)],
JRn
Proof. (1) First suppose that Ix  yl < 10· 2j . If we let ¢(u) = ¢(u 2 j (x  y)), then (as we vary x and y) ¢ runs over a bounded subset of C~(IRn). Further, ¢j = ¢j, so by the WBP
\Kj(x,y)\ If Ix 
I(T1/;J,¢j)\ < C2 nj < Cpj(xy).
=
y\ > 10· 2j , then ¢j Kj(x,y) =
J. J. Rn
Rn
and
1/;J
have disjoint supports, and so
¢j(xu)K(u,v)1/;j(vy)dudv.
Since 'ljJ has zero integral, this can be rewritten as
Kj(x,y) =
r r
¢j(xu)[K(u,v)K(u,y)]1/;j(vy)dudv.
JRn JRn
Because of the supports of ¢j and 1/;j, we must have that Ix  ul, Iv  yl < 2j and lu  yl ~ Ix  yl· This, together with inequality (5.13), which holds since K is a standard kernel, gives us that
C2 j8 IKj(x, y)\ < Ix _ yln+8 < CPj(x  y). (2) If Iw  xl > 2j then this inequality follows from (1). Therefore, we may assume that Iw  xl < 2j . If u is on the line segment between x and W then
pj(u  y) < C[pj(x  y)
+ pj(w 
y)].
Hence, (2) follows from the mean value theorem if
IVxKj(x,y)1 < C2 j pj(x  y). To prove this it suffices to note that
VxKj(x,y) = (T1/;j, V¢j) = 2j (T1/;j, (V¢)j) and then to argue as we did in (1). The same argument with V y gives us the analogous inequality in the second variable.
208
9. The T1 Theorem
(3) Formally, this identity is equivalent to T1 = O. To actually prove it, note that for R > 2j + 1 we have by Fubini's theorem that
r
J1Y1 '5,R
Kj(x, y) dy
(Th, ¢j),
=
where
h(u) = 2 jn
r
J1Y1 '5,R
'ljJ (u 2 . y) dy. J
This is zero if lui > R + 2j + 1 since supp('ljJ) C B(O,2), and it is zero if lui < R  2j + 1 since 'IjJ has zero integral and compact support. If R is sufficiently large then hand ¢j have disjoint supports, so we can evaluate (Th, ¢j) using (9.3). Therefore, by the standard estimates for K,
and this tends to zero as R tends to infinity. (4) Formally, this identity is equivalent to T*l = 0; we will use this fact to prove it. Arguing as we did above, for R > 2j we have that
r
J1xl'5,R
Kj(x, y) dx =
r
J1xl'5,R
Kj(x, y) dx  (T'ljJJ' 1) = ('ljJJ' T*h),
where
h(u) = 2 jn
r
J1xl'5,R
¢ (u . x) dx  l. 2J
Since ¢ has integral 1 and supp(¢) c B(O, 1), h(u) = 0 if lui:::; R  2j . For R sufficiently large, 'ljJj and h have disjoint supports, so by (9.3), and again since 'IjJ has zero integral and supp ('IjJ) c B (0, 2),
('ljJJ' T* h) =
r J
1vyl'5,2 j + 1
r J
[K(U,V)K(u,y)]h(U)2 jn'IjJ(v. Y )dUdV. 2J
1ul>R2 j
Therefore, by the standard estimates for K,
T* h) I < C2 jn I( nl'~ 'l'J' 
< C2j8

1,
r
.
IvYI'5,21+1
1,
.
lul>R21
IUIv Yyl8 In+8 du dv
du ,
J1ul>R/2 lul n +8
and the last term tends to zero as R tends to infinity.
o
4. Proof of the T1 theorem
209
To complete the proof we need to estimate the norm of TjT; as an operator on L2. The kernel of this operator is Aj,k(X, y)
=
r
JJRn
Kj(x, z)Kk(y, z) dz
and it satisfies two estimates. Lemma 9.16.
(1) For all x,
r IAj,k(X, y)1 dy < C2oljkl;
J'Rn
r
(2) for all y,
J'Rn
IAj,k(X, y)1 dx < C2oljkl.
Proof. By Lemma 9.15 IAj,k(X, y) I =
(9.8)
r r J'Rn
JJRn
K j (x, z) [Kk(Y, z) Kk(y, x)] dz Pj(x  z) min(l, 2 k lx  zl)[Pk(Z  y)
+ Pk(X 
y)] dz.
It is easy to see that
r Pk(Z 
J'Rn and that
r JJRn
y) dy
=C
pj(x  z) min(l, 2 k lx  zl) dx
< C2olkjl.
The desired inequalities now follow immedi;:ttely: we get (1) if we write the integral of (9.8) as
C
r pj(x z) min(l, 2 k lx JJRn
zl)
r [Pk(Z 
JJRn
y)
+ Pk(X 
y)] dy dz,
and we get (2) if we write the integral of (9.8) as
C
in
Pk(Z  Y) (/"" pj(X  z) min(l, Tklx 
+ c /"" Pk(X 
y)
(in
zll dX) dz
pj(X  z) min(l, Tklx  zl) dZ) dx.
o Given Lemma 9.16, it is simple to show that IITj T;IIL2,L2 < C2oljkl.
By the CauchySchwarz inequality, if f E S (IR. n ) , IITjT;
fll~ = J'Rrn J'Rrn Aj,k(X, y)f(y) dy 2 dx
210
9. The T1 Theorem
<
j"" (1. IAj,k(X, y)1 dY) (in IAj,k(X, 1{11If(y)1 dY) dx 2
< C228ljklllfll~· Essentially the same argument gives the same bound on the norm of Tj*Tk. Therefore, we can apply Cotlar's lemma to the RN'S, and this completes the proof of the case when T1 = T* 1 = O.
Case 2: Arbitrary operators T. Lemma 9.17. Given any function b E BMO, there exists a Calder6nZygmund operator L such that L1 = band L*l = O.
Assume Lemma 9.17 for a moment. Given an operator T which satisfies the conditions of the T1 theorem, suppose T1 = b1 and T* 1 = b2. Then there exist Calder6nZygmund operators Ll and L2 such that Ll1 = b1 , Li1 = 0, L21 = b2 and L21 = O. Then the operator T = T  Ll  L;' has the W BP and satisfies T1 = T*l = O. Therefore, by the previous argument T is bounded on L 2 and so T is as well. Proof of Lemma 9.17. Let cP and 'lj; be radial functions in S(JRn ) which are supported in the unit ball and such that cP is positive and has integral 1 and 'lj; has integral zero. Define the operator L by
Lf
=c
1
dt
00
o
'lj;t
* (('lj;t * b) (CPt * J)) , t
where c is a constant which we will fix below. (The operator L is called a paraproduct.) We will show that the kernel of L satisfies the standard estimates, that L is bounded on L2, and that L1 = band L*l = O. In order to be rigorous in the following calculations we ought to integrate between E and liE and then let E tend to O. However, we will omit these details.
(1) The size of the kernel. The kernel of L is
K(x, y)
=c
1l 00
o
lR.n
'lj;t(X  z)('lj;t
* b)(z)cpt(Z 
dt = y) dz
c
1
00
dt Kt(x, y).
tot
Fix z E JRn and let Q be a cube with center z and side length 2t, and let bQ be the average of b on Q. Then, since 'ljJ has zero average,
Therefore,
211
4. Proof of the T1 theorem
Since Kt(x, y)
= 0 if Ix  yl > 2t, we also have that IK,(x,y)1 <
Cllbll.cn
(1 + Ix ~ YI) ~n~'
It follows from this that
Cllbll*In . IK(x ,y )1 < Ixy By analogous arguments we can also show that
lV'xK,(x,y)1
+ IV'yK, (x, y)1 < Cllbll.cn~1
(1 + Ix ~ Ylrn~',
and so we have that
lV'xK(x,y)1
+ lV'yK(x,y)1 < Ix Cllbll* _ yln+1·
From these inequalities it follows that K satisfies (5.12), (5.13) and (5.14) with b = 1 and constant Cllblk
(2) Boundedness on L2. Let 9 E S(JRn) with
(Lf, g) =c r r=('ljJt * b)(x)(cpt ~h
IIgl12 < 1.
Then
* f(x))('ljJt *g(x)) dtdx t
(1"" 10= l,p, * b(x)I'I
J~n
1/'
r= l'ljJt * g(x) 12 dx dt) 1/2 t .
Jo
By Corollary 9.7 the first term is bounded by Cllbll*llfI12. In the second term, if we apply the Plancherel theorem to the inner integral and then exchange the order of integration and integrate in t, we get the bound Cllgl12. Hence, (Lf,g) < Cllbll*llfI121IgI12, so L is bounded on L2. (3) L1 = band L*l = o. For each t > 0 we have that
* (('ljJt * b) CPt * .)) * = CPt * (('ljJt * b) 'ljJt * .). Since 'ljJ has zero integral, 'ljJ * 1 = 0 and so it follows that L * 1 = o. Now let 9 E C;:Q. , Since for all x, CPt * l(x) = 1, we have that ('ljJt
(L1, g) =
c
r=· r 'ljJt * b(x)cpt J~n
Jo
* l(x)'ljJt * g(x) dx dt
= c r= r b(x)'ljJt * 'ljJt *g(x) dxdt.
h
~ t For this integral to equal (b, g), as desired, we need ='ljJt * 'ljJt * 9 dt = g, (9.9) c o t
1
t
212
9. The T1 Theorem
where the integral is understood to converge in HI. We omit the details. We note, however, that (9.9) determines the value of c which we left undefined above. If we take the Fourier transform we see that (9.9) holds in L2 if and only if for any ~ i= 0, c
tX) 1'IjJ(t~) 12 dt = l.
io
t
Since 'IjJ is radial, the integral does not depend on the value of c.
~,
and so this determines 0
5. Notes and further results 5.1. References.
Cotlar's lemma appeared for the first time in an article by M. Cotlar (A combinatorial inequality and its applications to L2 spaces, Rev. Mat. Cuyana 1 (1955),4155) for selfadjoint operators. The general case we give is due independently to M. Cotlar and E. M. Stein and appeared first in a paper by A. Knapp and E. M. Stein (Intertwining operators for semisimple groups, Ann. of Math. 93 (1971), 489578). Carleson measures and their characterization by Theorem 9.5 are due to L. Carleson (An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921930; and Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547559). Their relationship to BMO is due to C. Fefferman and E. M. Stein (HP spaces of several variables, Acta Math. 129 (1972), 137193). The T1 theorem is due to G. David and J. L. Journe (A boundedness criterion for generalized Calder6nZygmund operators, Ann. of Math. 120 (1984),371397). There are a number of proofs of this theorem; we followed the original, as further expanded in the thesis of G. David (Univ. Paris Sud, 1986). A much shorter proof is due to R. Coifman and Y. Meyer (A simple proof of a theorem by G. David and J.L. Journe on singular integral operators, Probability Theory and Harmonic Analysis, J. Chao and W. Woyczyilski, eds., pp. 6165, Marcel Dekker, New York, 1986); it also appeared in their article in Stein [16] and in the book by Torchinsky [19]. The original proof and yet another proof using wavelets appear in volume II of the book Ondelettes et Operateurs by Y. Meyer (Hermann, Paris, 1990; English translation in Y. Meyer and R. Coifman, Wavelets: Calder6nZygmund and Multilinear Operators, Cambridge Univ. Press, Cambridge, 1997). The operators in Lemma 9.17, the socalled paraproducts, were first introduced by R. Coifman and Y. Meyer [2]. For an overview including applications and their relation to the T1 theorem, see the monograph by M. Christ (Lectures on Singular Integral Operators, CBMS Regional Conference Series in Mathematics 77, Amer. Math. Soc., Providence, 1990).
5. Notes and further results
213
The version of the Tl theorem given by Stein [17] is very interesting. The W B P and the conditions on Tl and T* 1 in the statement of the theorem are replaced by a "restricted boundedness" condition for T and T*: IIT(¢x,R)112, IIT*(¢x,R)112 < CRn / 2 . The role of BMO appears in the proof: restricted boundedness and the conditions on the kernel are enough to define the action of the operator on constant functions and to realize it as a BMO function. But BMO does not appear in the statement of the theorem, and it only involves the space L2. 5.2. Calderon commutators and the Cauchy integral.
The operators given as applications of the Tl theorem in Section 3 were first studied by A. P. Calderon. He proved that the commutator T1 is bounded on L2 (Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 10921099) and proved Corollary 9.13 (Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 13241327). Also see his survey paper Commutators, singular integrals on Lipschitz curves and applications (Proceedings of the LC.M. (Helsinki, 1978), pp. 8596, Acad. Sci. Fennica, Helsinki, 1978). The boundedness on L2 of the remaining commutators was proved by R. Coifman and Y. Meyer (Commutateurs d'integrales singulieres et operateurs multilineaires, Ann. lnst. Fourier 28 (1978), 177202) with bounds worse than those in Corollary 9.12 (on the order of k!lIA'II~J These do not allow the series in the proof of Corollary 9.13 to be summed. The restriction of this corollary that IIA'IICXJ < E is not necessary: the result is true for any A such that A' E LCXJ. This was first proved by R. Coifman, A. McIntosh and Y. Meyer (L'integrale de Cauchy dejinit un operateur borne sur L2 pour les courbes lipschitziennes, Ann. of Math 116 (1982), 361387). G. David (Operateurs integraux singuliers sur certaines courbes du plan complexe, Ann. Sci. Ec. Norm. Sup. 17 (1984), 157189) obtained the general result starting from Corollary 9.13 by proving that if it is true for IIA'IICXJ < E then it is true for IIA'IICXJ < 19° E. A similar argument was developed independently by T. Murai (Boundedness of singular integral operators of Calderon type, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), 364367); see also his book which is cited below. Further, David gave a characterization of those curves on which the Cauchy integral defines a bounded operator on L2: they are precisely those curves such that any circle of radius r contains in its interior a piece of the curve of length at most Cr for C fixed. (These are usually referred to as AhlforsDavid curves.) In the book by T. Murai (A Real Variable Method for the Cauchy Transform and Analytic Capacity, Lecture Notes in Math. 1307, SpringerVerlag, Berlin, 1988) the first two chapters are devoted to the study of the L2 boundedness of the commutator T1 and the Cauchy integral. It contains several
9. The T1 Theorem
214
proofs of these results; of interest is the short proof for the Cauchy integral due to P. Jones and S. Semmes. This proof and a second one appear in a paper by these authors and R. Coifman (Two elementary proofs of the L2 boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), 553564); see also P. Jones (Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), pp. 2468, Lecture Notes in Math. 1384, SpringerVerlag, Berlin, 1989) and S. Semmes (Square function estimates and the T(b) theorem, Proc. Amer. Math. Soc. 110 (1990), 721726). A very elementary geometric proof of the general result was found by M. Melnikov and J. Verdera (A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices 1995, no. 7, 325331). 5.3. The bilinear Hilbert transform.
While studying the L2 boundedness of the commutator T 1 , Calderon showed via an argument analogous to the method of rotations (see Chapter 4, Section 3) that Tl is bounded if the family of bilinear singular integral operators Ha(f, g)(x)
= lim ('>0
r
lltl >(
f(x  t)g(x + at) dt
t
is a uniformly bounded map from L2(JR) x L=(JR) to L2(JR) for 1 < a < O. When a = 0 this operator reduces to Hf· 9 and when a = 1 to H(f· g), where H is the Hilbert transform. Therefore, the problem is to show that Ha is uniformly bounded for 1 < a < O. The operator HI is called the bilinear Hilb~rt transform. Calderon also posed the simpler problem of proving that HI is a bounded operator from L2 x L2 into Ll. For over thirty years no progress was made on these problems. Then in a pair of remarkable papers, M. Lacey and C. Thiele (LP estimates on the bilinear Hilbert transform for 2 < p < 00, Ann. of Math. 146 (1997), 693724, and On Calderon's conjecture, Ann. of Math. 149 (1999), 475496) proved the following result. Theorem 9.18. Given a E JR and exponents PI, P2 and P3 such that 1 < Pl,P2 < 00, P3 > 2/3 and l/Pl + 1/P2 = 1/P3, then
IIHa (J,g)llp3 < Ca ,Pl,P21Ifll p lllgll p 2' Theorem 9.18 establishes Calderon's conjecture for the bilinear Hilbert transform and shows that the operators H a , 1 < a < 0, are bounded.
5. Notes and further results
215
However, the constant C a ,2 ,00 is unbounded as a tends to 1 and 0, so this result does not complete Calderon's proof of the L2 boundedness of TI. But as this book was going to press, L. Grafakos and X. Li announced a proof that the constant is independent of a when 2 < PI, P2 < 00 and 1 < P3 < 2, and in all the cases covered by Theorem 9.18 if a is bounded away from 1. 5.4. The Tb theorem.
The Tl theorem does not give a better result for the Cauchy integral than Corollary 9.13 because we cannot directly calculate the action of the operator on the function 1. However, A. McIntosh and Y. Meyer (Algebres d'operateurs dejinis par des integrales singulieres, C. R. Acad. Sci. Paris 301 (1985), 395397) generalized the Tl theorem as follows. Let b be a function which is bounded below (more precisely, for some 8 > 0, Reb(y) > 8 for all y). IfTb = T*b = 0 and MbTMb has the WBP (where Mb is multiplication by b), then T is bounded on L2. If we let b(y) = 1 + iA' (y), then this is bounded below and one can show that the Cauchy integral of b is zero, so we can apply this result directly to the Cauchy integral. With this result as their starting point, G. David, J. L. Journe and S. Semmes (Operateurs de Calder6nZygmund, fonctions paraaccretives et interpolation, Rev. Mat. Iberoamericana 1 (4) (1985), 156) found a significant generalization of the T1 theorem, the socalled Tb theorem. To state it, we need several definitions. Given /L, 0 < /L < 1, define ct(IRn ) to be the space of functions of compact support such that
IIfll~ Given a function b, let bCt
=
sup If(lx )  f(y)1 x=/y x  YI~
= {bf
<
00.
: f E ct}.
A bounded function b : IR n 7 C is paraaccretive if there exists 8 > 0 such that for every x E IR n and r > 0 there exists a ball B = B (y, p) C B(x,r) with p > 6r which satisfies IbBI > 8. Theorem 9.19. Let bI and b2 be two paraaccretive functions on IR n and let
be an operator associated with a standard kernel K(x, y). Then T has a continuous extension to L2 if and only if Tb I E BMO, T*b 2 E BMO and Mb2TMbl has the WBP.
Because Theorem 9.19 is given in terms of the space ct instead of Cr;:, it can be generalized to spaces of homogeneous type (see Chapter 5, Section 6.4). There are other variants of this result, including defining Tb for
216
9. The T1 Theorem
functions b not in C= and a somewhat different version of.the weak boundedness property. For all of these results we refer to the articles cited above. See also M. Christ (A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601628), and the book by Meyer cited above.
Bibliography
[1] N. K. Bary, A Treatise on Trigonometric Series, translated by M. F. Mullins, Pergamon Press, New York, 1964. [2] R. Coifman and Y. Meyer, Au dele}, des operateurs pseudodifferentiels, Asterisque 57 (1979). [3] K. M. Davis and Y. C. Chang, Lectures on BochnerRiesz Means, London Math. Soc. Lecture Notes 114, Cambridge Univ. Press, Cambridge, 1987. [4] H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press, New York, 1972. [5] R. E. Edwards, Fourier Series: A Modern Introduction, 2nd ed., SpringerVerlag, New York, 1979. [6] J. GardaCuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland Math. Studies 116, North Holland, Amsterdam, 1985. [7] M. de Guzman, Real Variable Methods in Fourier Analysis, NorthHolland Math. Studies 46, NorthHolland, Amsterdam, 1981. [8] J. L. Journe, Calder6nZygmund Operators, PseudoDifferential Operators and the Cauchy Integral of Calder6n, Lecture Notes in Math. 994, SpringerVerlag, Berlin, 1983. [9] J. P. Kahane and P. G. LemarieRieusset, Fourier Series and Wavelets, Gordon and Breach, Amsterdam, 1995. [10] Y. Katznelson, An Introduction to Harmonic Analysis, Dover, New York, 1976. [11] P. Koosis, Introduction to Hp Spaces, 2nd ed., Cambridge Tracts in Mathematics, 115, Cambridge Univ. Press, Cambridge, 1998. [12] T. W. Korner, Fourier Analysis, Cambridge Univ. Press, Cambridge, 1988. [13] U. Neri, Singular Integrals, Lecture Notes in Math. 200, SpringerVerlag, Berlin, 1971. [14] W. Rudin, Real and Complex Analysis, 3rd ed., McGrawHill, New York, 1987. [15] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.

217
218
Bibliography
[16]
, ed., Beijing Lectures in Harmonic Analysis, PrincetonUniv. Press, Princeton, 1986.
[17]
, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993.
[18]
and G. Weiss, Introduction to Fourier Analysis in Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
[19] A. Torchinsky, Real Variable Methods in Harmonic Analysis, Academic Press, New York, 1986. [20] G. Weiss, "Harmonic Analysis," in Studies in Mathematics, vol. 3, 1. 1. Hirschman, Jr., ed., Math Assoc. of America, 1965, pp. 124178. [21] A. Zygmund, Trigonometric Series, vols. I and II, 2nd ed., Cambridge Univ. Press, London, 1959. , Integrales Singulieres, Lecture Notes in Math. 204, SpringerVerlag, Berlin,
[22] 1971.
Index
Al weights, 38, 134, 140 Ap weights, 133, 135 and BMO, 151 factorization of, 136, 150 reverse Holder inequality, 137, 151 two weight condition, 154 Aoo weights, 139, 149, 152 and doubling measures, 149 AbelPoisson summability, 10, 18, 25, 32 AhlforsDavid curves, 213 A; (strong Ap weights), 152 and strong maximal function, 153 factorization of, 153 approximation of the identity, 25, 31 atoms, 115 BanachSteinhaus theorem, 6 BesicovitchMorse covering lemma, 44 Besov spaces, 185 Bessel functions, 171 bilinear Hilbert transform, 214 BM (bounded mean oscillation), 117 and Ap weights, 151 and Carleson measures, 199 and commutators, 131 and interpolation, 121 dual of HI, 129 logarithmic growth, 123, 125 Bochner's theorem, 23 BochnerHecke formula, 85 BochnerRiesz multipliers, 47, 169, 187 and restriction theorems, 189 critical index of, 169
°
Co, 14, 131 C~,
xviii
C~o' 201 Ct,215 Calderon commutators, 100, 203, 213 conjecture, 214 Calderon Zygmund decomposition, 34 weighted, 150 operators, 98, 100 maximal, 102 norm inequalities, 99 singular integrals, 102 theorem, 91 Carleson measures, 197 and BMO, 199 and Poisson integrals, 198 CarlesonHunt theorem, 9, 18 Cauchy integral, 99, 109, 205, 213 Cesaro summability, 9, 18, 25, 32, 168 and Gibbs phenomenon, 21 Chebyshev inequality, 27 CoifmanFefferman inequality, 152 commutators, 131 conjugate function, 61, 155 conjugate Poisson kernel, 50, 62 convolution, xviii Fourier transform of, 11 Cotlar's inequality, 56, 102, 147 lemma, 114, 195 covering lemmas, 31, 44 BesicovitchMorse, 44 Vitalitype, 44

219
Index
220
decreasing rearrangement, 40 weighted, 152 differential operators, 80, 83, 88, 113 Dini's criterion, 3 Dinitype condition, 93, 107 Dirichlet criterion, 2 kernel, 3, 18 problem, 11, 19 distribution function, 28 distributions, xviii homogeneous, 71 tempered, xviii, 13 doubling measures, 44, 109 and Aoo weights, 149 du BoisReymond theorem, 6 dyadic cubes, 33 dyadic spherical maximal function, 178 elliptic operator, 82, 114 extrapolation theorem, 141 Fejer kernel, 9, 18 Fourier series, 2, 61 almost everywhere convergence, 9 coefficients, 2 uniqueness of, 10 conjugate, 62 convergence in norm, 8, 62, 68 divergence, 6, 9 Gibbs phenomenon, 21 multiple, 20, 67, 112 multipliers of, 67 of LI functions, 9 of L2 functions, 9 of continuous functions, 6 partial sum, 2, 67 pointwise convergence, 2 summability methods, 9, 25, 32 Fourier transform, 11 almost everywhere convergence, 18 eigenfunctions, 22, 85 inversion formula, 13 of L2 functions, 15 of LP functions, 16 of finite measures, 14, 23 of tempered distributions, 14 partial sum operator, 17, 59, 67 properties, 11 restriction theorems, 188 and BochnerRiesz multipliers, 189 summability methods, 17, 188 fractional integral operators, 88 and BMO, 130 and HI, 130 and fractional maximal function, 90, 130
GaussWeierstrass kernel, 19 summability, 18, 25, 32 Gibbs phenomenon, 21 goodA inequality, 121 weighted, 144, 152 gradient condition, 91
HI, 116, 127 and interpolation, 129 H~t, atomic HI, 116 HP (Hardy spaces), 126 and grand maximal function, 127 and Lusin area integral, 128 atomic decomposition, 128 dual spaces, 129 nontangential maximal function, 127 on spaces of homogeneous type, 128 Hardy operator, 39 HausdorffYoung inequality, 16,21,41 heat equation, 19 Hedberg's inequality, 89 HelsonSzego condition, 148 Hermite functions, 22 Hilbert transform, 49, 51 along a parabola, 179 and BMO, 120 and HI, 116 and multipliers, 58 bilinear, 214 directional, 74, 86 L log L estimates, 66 maximal, 56, 76 norm inequalities, 52, 197 of L1 functions, 55, 65 of characteristic functions, 54, 68 pointwise convergence, 55 size of constants, 54, 64 truncated, 55, 64, 197 weighted inequalities, 155 Hormander condition, 91 for Calder6nZygmund operators, 99 for vectorvalued singular integrals, 106 on spaces of homogeneous type, 109 multiplier theorem, 164, 190 interpolation analytic families of operators, 22 and BMO, 121 and HI, 129 Marcinkiewicz, 29 and LP,q (Lorentz spaces), 41 size of constants, 30 RieszThorin, 16
Index
JohnNirenberg inequality, 123 converse, 125 Jordan's criterion, 3 Kakeya maximal function, 46 Kakeya set, 47 kernel conjugate Poisson, 50, 62 Dirichlet, 3, 18 Fejer, 9, 18 GaussWeierstrass, 19 Poisson, 11, 19, 49, 61 standard, 99 Kolmogorov and divergent Fourier series, 9 lemma, 102 theorem, 52, 62
LP, xvii Lco, xviii LP(B), 105 LP(w) (weighted LP), 133 L~, 90, 163
LP,q (Lorentz spaces), 41, 112 L~, 163
Laplacian (6.), 80, 88 Lebesgue differentiation theorem, 36, 46 number, 7 point, 36 Lipschitz spaces, 114, 129 LittlewoodPaley theory, 159, 185 weighted, 186 localization principle, 4 Lusin area integral, 128 Marcinkiewicz interpolation and LP,q (Lorentz spaces), 41 interpolation theorem, 29 multiplier theorem, 166, 186, 190 maximal functions along a parabola, 179 directional, 74, 87 dyadic, 33 and sharp maximal function, 121, 144 dyadic spherical, 178 fractional, 89, 130 grand, and HP spaces, 127 HardyLittlewood, 30, 133 L log L estimates, 37, 42 and approximations of the identity, 31 and dyadic maximal function, 35 as vectorvalued singular integrals, 111 noncentered, 30, 133 norm inequalities, 31, 35 on BMO, 129
221
reverse (1, 1) inequality, 42 size of constants, 43, 75 weighted norm inequalities, 37, 135, 137, 152, 154 weights involving, 37, 134, 140, 156 Kakeya,46 nontangential, 45 and HP spaces, 127 onesided, 40 rough, 74, 87 sharp, 117, 121, 130, 144 and singular integrals, 143, 152 spherical, 191 weighted norm inequalities, 192 strong, 45 L log L estimates, 46 weighted norm inequalities, 152 weights involving, 153 with different measures, 44, 153 maximal operator of a family of linear operators, 27, 56, 75, 147, 152 method of rotations, 74, 84, 86, 179, 214 Minkowski's integral inequality, xviii multiplier theorems Hormander, 164, 190 Marcinkiewicz, 166, 186, 190 multipliers, 58, 66, 163 BochnerRiesz, 47, 169, 187 and restriction theorems, 189 critical index of, 169 of Fourier series, 67 weighted norm inequalities, 164, 190 nonisotropic dilations, 108 nontangential approach regions, 45 nonhomogeneous spaces, 109 operator algebras, 80, 86 Orlicz spaces, 42, 66 orthogonality relations, 2 paraaccretive functions, 215 parabolic operators, 108, 178 paraproducts, 210 Plancherel theorem, 15 Poisson integrals, 19, 49, 70 and HP functions, 126 and Carleson measures, 198 nontangential maximal function, 45, 127 kernel, 11, 19, 49, 61 conjugate, 50, 62 summation formula, 20 potential logarithmic, 70 Newtonian, 70
Index
222
principal value of l/x, 50 of Ixlnit, 97 pseudodifferential operators, 85, 113 pseudometric, 109 Rademacher functions, 177 restriction theorems, 188 and BochnerRiesz multipliers, 189 reverse Holder inequality, 137, 151 reverse Jensen inequality, 150 Riemann localization principle, 4 RiemannLebesgue lemma, 4, 11 Riesz potential, see also fractional integral operators theorem, 52, 62 transforms, 76, 110, 150 and HI, 116, 127 and differential operators, 81 RieszThorin interpolation theorem, 16 rising sun lemma, 40 S, xviii, 12 S',13 Schwartz class, xviii, 12 singular integrals, 69, 91 and BMO, 118 and HI, 116 and VMO, 131 and LittlewoodPaley theory, 172, 192 and pseudodifferential operators, 113 and the Hilbert transform, 70 as convolution with tempered distributions, 69 maximal, 75, 147, 152 norm inequalities, 75, 79, 84, 87, 91, 93, 97, 106, 172, 176, 192, 203 on curves, 193 size of constants, 110 strongly singular, 112 truncated, 55, 94, 110 vectorvalued, 106, 157 and maximal functions, 111 weighted norm inequalities, 144, 145, 147, 150, 152, 155, 159, 192 with even kernel, 77 with odd kernel, 75 with variable kernel, 84 Sobolev embedding theorem, 90 space, 90, 114, 163 BMO and exponential integrability, 130 spaces of homogeneous type, 109 and HP spaces, 128 spherical harmonics, 85
square functions, 176 standard kernel, 99 sublinear oper~tor, 28 summability methods AbelPoisson, 10, 18, 25, 32 Cesaro, 9, 18, 25, 32, 168 and Gibbs phenomenon, 21 from BochnerRiesz multipliers, 188 GaussWeierstrass, 18, 25, 32 T1 theorem, 203 Tb theorem, 215 tangential approach regions, 45 TomasStein theorem, 189 translation invariant operators, 66 TriebelLizorkin spaces, 185
uniform boundedness principle, 6 Van der Corput's lemma, 183 Vitalitype covering lemma, 44 V MO (vanishing mean oscillation), 131 and commutators, 132 and singular integrals, 131
WBP (weak boundedness property), 202 weak (p, q) inequalities, 26 almost everywhere convergence, 27 weights and AI, 38, 134, 140 and Aoo, 139, 149, 152 and Ap, 133, 135 and A;, 152 Young's inequality, 17, 22