TOPICS IN ANALYSIS AND
ITS APPLICATIONS
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TOPICS IN ANALYSIS AND
ITS APPLICATIONS SELECTED THESES
Editor
R Coifman Department of Mathematics Yale University USA
W g World Scientific wk
Singapore 'New Jersey London* Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 91280S USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publtcatlon Data Topics in analysis and its applications : selected theses / edited by R. Coifman. p. cm. Includes bibliographical references. ISBN 9810240937 ISBN 9810240945 (pbk) (alk. paper) 1. Harmonic analysis. I. Coifman, Ronald R. (Ronald Raphael) QA403.T62 2000 515\2433--dc21
00-021464
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Printed in Singapore.
V
Introduction
The five selected theses in analysis, by A. Gilbert, N. Saito, W. Schlag, T. Tao and C. Thiele, cover a broad spectrum of modern Harmonic Analysis from Littlewood-Paley theory (wavelets) to subtle interactions of geometry and Fourier oscillations. The common theme through the theses involves in tricate local Fourier (or multiscale) decompositions of functions and operators to account for cumulative properties involving size or structure. Anna Gilbert's thesis addresses the question of homogenization for partial differential equations and transitions from fine scale to course scales. The purpose is to provide effective equations to describe the solution of a PDE in mixed media. This is achieved by proving that solutions of governing PDE in unmixed media converge weakly to solutions of some effective equation as the medium is randomly mixed. By setting the problem up as a sequence of transitions from solutions of equations on a fine (micro) scale and by deriving equations for the averages of the solution on a coarser scale, A. Gilbert provides a general method for direct numerical analysis for the transitions to macroscopic scales. She man ages, for various examples, to compute analytically the effective equations with surprising results. The main point is that effective transitions from a microscopic regime to a coarse regime can be tracked without having to assume that we have random mixing. It is enough to describe transitions between neighboring scales. The use of wavelets in this work provides the algebraic structure needed to convert Littlewood-Paley theory into a systematic numerical tool, by providing a detailed account of transitions from high frequencies in the fine structured equations to low frequencies in the effective coarse structure. Terence Tao in his thesis develops tools for the analysis of various classes of singular integral operators. In the first part he shows that the spherical Fourier restriction property of Thomas-Stein implies a corresponding weak type estimate for Bochner-Riesz multipliers. (This result is then generalized for uniformly elliptic first-order pseudo-differential operators.) Here again the
vi Introduction
important contribution is in the method of decomposition of this oscillatory operator, which allows for a precise control of its action on the corresponding Calderon-Zygmund decomposition of the function. In the next part of his thesis, Tao shows that the numerically natural way for truncating a wavelet expansion is a.e. convergent. (The same statement is false by Fourier Series as shown by T. Komer.) In the last part he generalizes a weak type estimate of A. Seeger for even singular integrals with ft in L log L to the homogeneous group setting. Here again the crucial issue is to obtain a good decomposition of the operator. N. Saito's thesis is concerned with the decomposition and analysis of func tions for extraction of "structures and features" useful for discrimination and regression. Here again the main point is to represent a function as a com bination of local Fourier series, where the local expansion is chosen so that the large coefficients will be useful for characterizing the "geometry" of the function: This can be achieved by finding an orthogonal local sine basis which minimizes the description length of the function. Similarly a local basis can be chosen to maximize discrimination between two classes of signals (functions). This thesis in applied mathematics develops a variety of analysis tools which are then applied to the analysis of acoustic ground signatures for identification and discrimination. The time frequency representations used here to decompose functions are also the fundamental tool in Thiele's thesis concerning the bilinear Hilbert transform of Calderon. Wilhelm Schlage addresses the problem of estimating the maximal circular averages operator by geometric combinatorial methods. Ever since E. Stein discovered, by using the Fourier transform, that the maximal spherical operator is bounded on some l? in R 2 for n > 3 and Bourgain proved the result in R 2 , there have been numerous attempts to prove these results by purely geometric means in which careful combinatorial coverings by annuli lead to geometric measure theoretic estimates. Schlage extends the methods of Bourgain, Kolasa-Wolff to extend the known range of (Lp,Lq) estimates. Again the main point is the subtle in teraction between the geometry of circles and Fourier analysis estimates. The maximal operator is a generalized Fourier integral operator requiring consider able geometric insight to obtain the appropriate decomposition of the operator. C. Thiele in his work starts by recasting the a.e. convergence for Walsh series in terms of decomposition theorems in the Walsh multiscale phase plane.
Introduction
vii
This work builds on the insight of Carleson, Hunt, Fefferman, and serves as a warmup to the attack on the Walsh model for the bilinear Hilbert transform, which together with Michael Lacey, leads to the proof of Calderon's conjecture on the boundedness of the bilinear Hilbert transform. (For which they were awarded the Salem Prize.) Here again we see the same main ingredients. The functions are broken up into a multiscale Fourier series in a way that controls and simplifies the bilinear (and trilinear) interactions. These decompositions are obtained by peeling off layers of large Fourier coefficients at different scales so as to control the operator. The methodology developed in this thesis and subsequent work with M. Lacey have converted Carleson's "tour de force" in proving the a.e. convergence of Fourier series into a general tool of Harmonic analysis. It is quite clear that the main contribution of all five Ph.D's is in providing deep organizational insight to complex interactions between localized Fourier components of the operators or functions being studied. The theorems and results are important, but it is the various methods of decomposition which are the enduring life blood of analysis. These tools have a much broader range of applicability, from linear and nonlinear partial differential equations to fast numerical algorithms, and are bound to have a lasting impact. These theses are beautifully written and are either self-contained or well referenced. They provide a valuable advanced textbook for a "topics" course in Harmonic analysis and its applications.
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be
Contents
Introduction
v
l? —¥ Lq Estimates for the Circular Maximal Function Wilhelm Schlag
1
Three Regularity Results in Harmonic Analysis Terry Too
61
Time-Frequency Analysis in the Discrete Phase Plane Christoph Martin Thiele
99
Multiresolution Homogenization Schemes for Differential Equations and Applications Anna C. Gilbert
153
Local Feature Extraction and Its Applications Using a Library of Bases Naoki Saito
269
1
£ p _>. L lP(Rd) for the optimal range of p's provided d > 3. The case d = 2 was settled by Bourgain. 1 It is easy to see that maximal functions of type (1.1) can never be bounded on any IP with p < oo if we replace spheres by boundaries of cubes, say. Indeed, let Mf{x)
= sup f
|/(*-*y)|d/x(y),
K t < 2 78[_l,l]
where dfj. is surface measure on d[—1, l]d. Choose / : R d -» R with the following properties: / > 0, f(x', 0) = oo for all x' = ( x i , . . . , I<J_I), / \f\p dx < oo for any finite p. With this choice of / it is clear that Ai(x',x < j) = oo for all x' € R d _ 1 , Xd G (1,2). More generally, the same example shows that maximal averages over any hypersurface containing a piece of a plane can never be bounded on IP for finite p. Typically, such hypersurfaces are ruled out by assuming nonvanishing Gaussian curvature. It turns out that this condition plays a crucial role in the analysis of M, and also in other problems in harmonic analysis, cf. Ref. 20. The reason for this is the following well-known result (which we state for spheres rather than general surfaces). Proposition 1.1.1. The Fourier transform of the surface measure da of the unit sphere 5 d _ 1 has the following asymptotic expansion: da(£) = e 2 -l«lu, + (K|) + e- 2 -KI W _(K|)
(1-2)
with smooth functions u>± on R + satisfying d
"
M 1. By stationary phase, the main contributions to the integral in (1.3) will come from the critical points of the phase function : 5 d _ 1 -► R denned by (x) = x • £. Clearly, those critical points are given by the points with normal parallel to v, i.e. x = ±v. Further more, by the nonvanishing of the Gaussian curvature, these critical points are nondegenerate. Hence stationary phase implies that the main contributions to (1.3) come from A - 1 / 2 neighborhoods of ±v, more precisely da{Xv) = Cle2iriX\-(n-1)/2
+ c 2 e- 2,r/ 2 + lower order terms.
Finally, the full statement (1.2) of the proposition follows by a more detailed analysis involving the parameter v G 5 d _ 1 . We refer the reader to Theo rem 7.7.14 of Ref. 6 or Theorem 1.2.1 of Ref. 16, for details. □ Following Stein, we prove the L2 boundedness of Mf(x)= sup / KK2JS''-1
\f(x-ty)\da(y)
(1.4)
in dimensions d > 3 using (1.2), cf. Refs. 18 and 20. Proposition 1.1.2. Suppose d > 3. Then WMfh < C\\fh for all f € Cc(Rd) ( = continuous functions with compact support).
(1-5)
Proof. Let Atf(x)
= f d 1 f(x- ty) da{y) = ( / * dat){x), Js -
where dat is the normalized surface measure on tSd~l. / > 0. Then f2\d
We may assume that
I
sup Atf(x) •*
Since dat{x) = t - ( d _ 1 W ( x / t ) and thus dat(0
i^
r
(1.8)
= <M*£)> (1.2) implies that
< C
(1.9)
provided d > 3. The proposition follows from (1.6)-(1.9).
□
R e m a r k s , (a) The operator M is not bounded on L 2 if d = 2, cf. Stein's example (1.11) below. (b) Estimate (1.5) also holds for the global maximal function M. Indeed, given suitable inequalities for M one can pass to the corresponding bounds on M by scaling and Littlewood-Paley theory. This is done in detail in Sec. 3.1 below. In this Introduction we shall consider only «M. (c) It is possible to extend Proposition 1.1.2 to general / G L2. Some care has to be taken in denning M, see, e.g., Sec. 1.3. It turns out that Proposition 1.1.2 is not optimal, in so far as M is bounded on Lp(Rd) for some p < 2 provided that d > 3. Indeed, we have the following result, which is due to Stein 18 in dimension d > 3 and Bourgain 1 in dimen sion 2. Theorem 1.1.1. Suppose d>2, d/(d - 1) < p < oo. Then (1.10)
\\Mf\\p(R») for any f £ C C (R 2 ). Moreover, (1.12) cannot hold for any (l/p,l/q) exterior of T.
(i.i2) in the
In view of Bourgain's theorem, (1.12) also holds if ( l / p , l/q) lies on the segment connecting (0, 0) with (1/2, 1/2). The optimality statement is easy. Indeed, for small S > 0, let /o = X[i-* 3 follows from (1.15). Indeed, by Proposition 1.1.1 spherical means can be written as Atf(x)
= f
e2"(*>dxdt < C e ||/||£, where 2 < p < o o , £ > 0 and
(R3),
(1.18)
8
W. Schlag
11(1-1) a(p)=
2V2 1
2 0, then \E\ > 0. In other words, the union of a family of circles has positive (planar) measure if their centers form a set of positive measures. Proof. This is a simple consequence of Bourgain's circular maximal theorem. Indeed, assume that U € R2 is open. Then there is an increasing sequence of non-negative functions / „ € C C (R 2 ) so that sup n / „ = \U- Fix a p e (2, oo). By monotone convergence and Theorem 1.1.1 = C\U\1^.
\\Mxu\\P 0. We could then conclude from the previous inequality that \\MXE\\P
=
0.
However, this contradicts | F | > 0.
□
One might ask whether the conclusion of Theorem 1.3.1 will still hold under the assumption that dim(F) > CQ for some co < 2. The following result of Talagrand 21 shows that Co has to be at least one. Proposition 1.3.1. There exist E and F such that H^F) > 0 but \E\ = 0. On the other hand, Wolff22 has shown recently that for F as in the propo sition, E "barely fails to have positive measure". More precisely he showed Theorem 1.3.2. IfHx(F)
> 0, then dim(E) = 2.
His proof is based in part on an argument from combinatorial geometry that was developed in Ref. 4 to obtain bounds on the number of incidences between n spheres and m points in R 3 . We do not know whether it is possible to deduce CQ = 1 from Ref. 22. Rather, we will show that CQ = 1 would be a consequence of the sharp local
10
W. Schlag
smoothing conjecture from the previous section. This connection, which relies on the theory of capacities, was brought to our attention by Thomas Wolff. For the definition of capacity as well as the connection between Hausdorff measure and capacity we refer the reader to the Appendix. P r o p o s i t i o n 1.3.2. Given that the sharp local smoothing conjecture is true, suppose that dim(F) > 1. Then \E\ > 0. Proof. Firstly, we may assume that E is compact and that
Hl(EnC(x,rx))>T>0 for all x € F with 7 fixed (since H1 is a regular outer measure, see Ref. 5). Since dim(F) > 1 there is an e > 0 such that H1+e{F) > 0. Fix e to be that number. Assume that the proposition fails. Then there is a sequence fj of non-negative functions in Co°(R 2 ) such that fj = 1 on a neighborhood of E snd 11/jlU -> 0 as j -> 00. Pick a cutoff function n € C Q ° ( R ) such that 77 = 1 on (1,2). Define Uj(x,r) = (dar *
fj){x)n{r).
Then Uj > 7/2 on some neighborhood of F' = {(x,r x ) : x € F}. If the sharp local smoothing conjecture is correct, then
Therefore, by the definition of capacity (see the Appendix) C 4 ,i/ 2 -«/8(F') 00, this would imply that C4,i/2-e/s(F')
= 0.
(1.21)
However, by the proposition in the Appendix we conclude from (1.21) that 0=
ft3-4(l/2-e/8)+e/2(F/)
which contradicts the choice of e.
_ %l+e{F')
>
H1+C\F) □
IS —► Lq Estimates for the Circular Maximal Function
11
2. The Main Argument 2.1. A reformulation
of the main
theorem
For the combinatorial argument below it is convenient to consider maximal averages over thin annuli rather than circles. More precisely, let 0 < S < 1/2, 1 < r < 2, and define for / G 5 C(x,r) = { x € R 2 :
|z-y|=r},
Cs(x, r) = {x 6 R 2 : Msf(x)= sup
,
r ( l - 6) < \x - y\ < r ( l + 6)} , ,
J
/
f(y)dy
where day is the normalized surface measure on rS1. Unless stated to the contrary, we shall be dealing only with functions defined on R 2 . We shall write < to denote < up to an absolute constant. Similarly with > and ~ . For any measure v on R d , we let v\(x) = X~di/(X~1x). First we will consider some examples, two of which have already appeared in Sec. 1.1. Examples. 1. Let / = Xc,(o i)- Then Msf(x)
~ 1 on |x| < 6. Hence ||/|| p ~ S1^ and
IIAWII, > * 2/ '2. Let / = Xfl. where R is the rectangle centered at 0 with dimensions 6 x 61'2. Then Msf(x) ~ <J1/2 provided that |x!| ~ 1 and |x 2 | < Sl/2. Hence 3/ 2p ||/|| p ~ and \\Msf\\q ~ S^1+1^2. 3. Let / = XB(O,S)- Then Msf(x) ~ 6 for |x| ~ 1 and thus ||/|| p ~ S2/p, \\Msf\\q^S.' Then 4. Let / ( x ) = (|1 - |x|| +6)-V2XB(oMx)-
s \x\-V2 Msf(x) > ,log^-r X
if 26 < |x| < 1
(2.1)
To see this write / as
f~5-1'2
E
2
j/2
Xcwo,i+(2>-i)«) •
(2.2)
K2K6-
Taking the average of / over the annulus Cs(x, 1 + |x|) and considering the contribution of each dyadic shell in (2.2) separately yields (2.1). Hence ||/||2~|log«5|1/2and||^/||2>|log*i}|>£|C,|.
(2.13)
The combinatorial method attempts to bound /i from above, typically in terms of A, M and 8. Since
M|£| > /
$ = 2 |{CJ: $ < M}| > AM 0, 1 < q < 00, 1 < p < 00, and all f G L 1 n L°°
l|AWll,;S*-l/llr. 7%en
(2-15)
L? -¥ Lq Estimates for the Circular Maximal Function
At A}, it follows that |{^X£>A}|<M«52. In view of (2.14), i.e. \E\ > p.~1XM5, and our assumption on /i we conclude that the right-hand side of (2.17) is > S-"'A1/pX-l(A-1X1+aM1-0S)1/p
~ (M62)1'* .
To prove the second statement, we distinguish two cases. First assume that \E1\ =
\{E:^>lx}\ + 1 }, and all 0 < t < 1, x0 e R 2 l l ^ / | | ^ ( f l ( I O , 0 ) < Cot1/2||/||L>(R>) •
(2-24)
Proof. We may assume that x 0 = 0. Choose cutoff functions %j) G C Q ° ( R 2 ) with V = 1 on B(0, 1), 77 G C*£°(l/2, 4) so that r j = l o n (1, 2), and G 5 such that supp( = 1 on {1/2 < |£| < 2}. Define Aif(x)
= 1>(t-lx)r,(r) [
e 2 ™
|^|- / (l + | r - ^ | | ) - 3 r d r
~
2'J2[ * > ^ K - 2 ) 3 / 6 . Thus N
N
^2ni = Y,(ni-2) + 2N i=l
t=l N
N
i-2)3) Vj=i
1/3
N2/3 + 2N
/ 1/3
N*'3 + N
< #5/3
Since card{(i,j): CitCj are tangent} = £ ) i = 1 "t, we are done.
□
26
W. Schlag
This proof is just a special case of a well-known argument that provides upper bounds for the maximal number of edges in a bipartite digraph with m edges and n sinks containing no Ks. Here ei(x, r) = — \Xi -
rsgn(ri - r ) . X\
Then Ns(S) < ( | )
A" 3
for any 0 < S < e. Remark. It is easy to see that the bound on Ng(S) can be attained. Proof. Let fi = {(x,r) € R2 x (1,2): \x-xj\>3e,r?ri, for i^j
|ei(x,r) - ej(x,r)\
,i,j = 1,2,3} 3
and F: fi -> R be denned by F(x,r) = ( | x i - * | - | r i - r | ) | L 1 . It is easy to see that the Jacobian JF of F satisfies
>A
IP —► Lq Estimates for the Circular Maximal Function
27
JF ~ |ei - e2\\ei - e 3 ||e 2 - e 3 | > A3 . Since card(F _ 1 (p)) < CQ for some absolute constant Co and all p € R 3 , we conclude that \F-l(B(0,2e))\<e3X-3. According to the definition of S there exists a function r : S —► (1,2) such that for every x € 5 we have \F(x,r(x))\ < e. Then clearly {(x,r): i e 5 , |r - r(x)| < £> C F - 1 (fl(0,2e)) Ui(x,r):xeSn
( J fl(x,-,3e), |r - r(z)| < e - |
and thus | 5 | < £2A~3.
D
The following lemma contains bounds on the diameter and the area of Cs(x, r)C\Cs(y, s). In various forms it appears in several papers on this subject, see, e.g. Refs. 1, 8, 9 and 15. Since the exact version we use here does not seem to be contained explicitly in any of these references, we provide a proof for the reader's convenience. Let A = max(||x-y|-|r-5||>($). L e m m a 2.4.2. Suppose x, y € R 2 , x jt yt \x — y\ < 1/2, and r,s € (1,2), r =fi s, 0 < 6 < 1. Then there is an absolute constant A such that (a) Cs(x, r) n Cs(y,s) is contained in a 6 neighborhood of an arc of length < Ay/A/\x — y\ centered at the point x — rsgn(r — s) i*"^. (b) The area of intersection satisfies \Cs(x,r)nCs(y,s)\ Li Estimates for the Circular Maximal Function
i™ ~ X ? Xc< ~ card(5'-e)^ ~ M ^'
29
(2 46)
'
On the other hand, by Lemma 2.3.3 On the other hand, by Lemma 2.3.3
Thus
Hence, if M(i)
(i)
,
and M 1 / 2 ,
(2.49)
then
= Bs\~3'2M1'2. Here we have used (2.45) and then (2.49) in line (2.50).
(2.50)
30
W. Schlag
Case 2:
(2.51)
A>100A(-J
Following Ref. 8 we let Q = {(j, *i,*2»*3): 1 < J1 < A/,ii,t2 > t3 6 5j e and the distance between any two of the sets Cj n Cj,, Cj D C i2 , C, D Q , is at least A/20}. (2.52) Suppose (j,h, 12,13) G Q. Then Lemma 2.4.2 implies that any two of ei = xj - Tj sgn(r,-
-n)-^-
\Xj -
Xi\
for i = ii, %2, 13 are separated by a distance A/20. Indeed, by that lemma, ei is the center of Cj n Cj and in view of (2.36), for any i € Sf e 7 inn Cj) C J )'{r)\dr J\x\ or equivalently
f
1>(x) = / (XB)r(x)p(r)dr, Jo where B is the unit ball in R 2 . Note that /
ip(x) dx = I
JK*
ip(r)r dr •
p(r) dr =
JO
JO
Let / € S. Then sup \dat *{<j>6*f)\((/i)2k)2fc>0
j>k
2-ka(M(fj)2>h-> fc>0
j>fc
9\l/9
(3.2)
36
W. Schlag
The first term in (3.2) is bounded by Lemma 3.1.2. Assume first that 0 < 0. Let s = max(2,p). Using the inequalities of Young and Littlewood-Paley, we can then estimate the second term as follows. 9 \ 1/91
2_fco
(£|E "*>0'j>k
k
(^)2 )2->D 9 \ 1/9
= (E(E 2 " fco ii(^(/^)2-ii,) 9 ) 2
, 9\ V?
w
s(E(£ °- ii/iiipY) *>0 ^j>k
'
'
^(Ei^iip) \1/2II
11/ N\
j
/
(3.3)
Up
If f3 > 0 we compute, starting in line (3.3) above,
/ I
(E E
2_fca
0 fc>0 j>k I i>Jb
| 9\\9\:1/911
-
(^(/i)2*) 2 - ) 'I ''
2 i£2 +e)
(E ^(E " ^k>0
S>*
^
'U 9\l/9
9
ii/iiip) ) 9 \ 1/9
,
Atfa-i
(3.17)
(3.18)
L^R*)
for all 1 < q < oo. Since
M?/l(2-j0d£
= [
JR*
= /+ + / _ . It suffices to consider J_. Introducing polar coordinates yields
42
W. Schlag
J_ = 2jd f Jo
e2ni2i^-t)ru^{2j\x\r)w-(2jtr)4>(r)rd-1dr
+ 2jd j Jo
X
e- 2 i r i 2 i ( | l l + t ) r w 0 -(2 J >|r)a;-(2 J 'tr)^(r)r < J - 1 dr
= A + B. Consider the first integral. |A| < CNVd(V\\x\
- t\)~N j f ° I (J^j
(r)rd-1} (2 , >|) f c (l + 2 J >|r)-( < J - 1 )/ 2 - f c (2 J 't)'
fc+/<w'1/4 x (1 + 2 ^ r ) - ( d - 1 ) / 2 - a - ' dr < Ca,N2>d{2j\\x\
- t\)~N{\ + 2 j |i|)-( < J - 1 )/ 2 2-^d-^2+^
.
Hence
\A\ 0
C {R2: 2*~l < |£| < 2 J + 1 }
I|M 1 / 2 /IIL>O(R*) < C , y | | / | | L i „ T ( R a ) •
(3.21)
Proof. By Eq. (4) in Ref. 18,
-***& = WZ\ f-VtfW1 1 (") Jo
- »2)a-l»d*
(3.22)
provided that 8?a > 0. Let a = e+ir. In view of (3.22), Theorem 3.1.1 implies that ||Ala/||5 0. Then we can choose C\ sufficiently large (depending only on r) so that a satisfies (3.24). For suitably small r depending on 0 and r) we conclude from (3.30) that = \-aM*,
H < X-V+I/V+WMW-""^^™
(3.32)
where 1 < a < 2 and a —¥ 1 astj —*0,a—t2asT] —> 1 and 0 —> 0. Moreover, 1
-±Z = 6 + 21-2/-'1>6 1-/3 1 + 2/3 + 77
and
I±S - 6 1-/J
46
W. Schlag
as r\ —► 1 — 2/3. Thus, according to Lemma 2.2.1, (3.32) corresponds to weak type p -> q estimates with 2 < p < 3 and q > 6. Case 2:
A>C0(-J
•
(3.33)
We rewrite (3.33) as
^(f(flSi)V. With Q defined by (2.52), we infer from Lemma 2.4.1 that card(Q) < ( j ) r 3 M m i n ( l , J^j
.
(3.35)
The minimum occurs on the right-hand side because for a given choice of xit we must have |Zi2 - x», | < |x t > 2e and ||»" —rj| — | x - i j | | <e imply that s g n ( r - T j ) = o~j. Consider first the case where o\ = /?T-8T£>/3T. If <Ti ^ CT2, then by a similar calculation t 2 a 2 > /3r. We conclude that
W -¥ L* Estimates for the Circular Maximal Function 49 JF
~^T
oaQnF-\D(0,e)).
By the coarea formula,
H1(F-1(y)nn)dy=
/ JD(0,c)
f
JF(x,r)dxdr,
yflnF-'(D(0,£))
eH>^\SlnF-HD(0,e))\ which implies \Projv(SinF-l(D(0,6)))\<e-^.
D
For example, set /? = e and t = T in (3.42). Then (3.42) says that the total number of circles in C^1 n Cft3 is no larger than the maximal number of circles in C^tl that pass through one of the points C\ n C 2 . To estimate \Cftl n C ^ 2 | in those cases where Lemma 3.4.1 does not apply we will use the following observation. Roughly speaking, it says that if Cj = C(XJ, 3/4) are internally tangent to C(0,1) for j = 1,2 with the points of tangency being far apart, then C\ and C2 cross each other. Lemma 3.4.2. Let Cj = C{XJ,TJ) for j = 0,1,2. Suppose A(C0,Cj) and \r0 — r,-| < 4pj, pj Ao.m±M£l±£lL V P1P2
(3.43)
for some sufficiently large constant AQ. Then A(Ci, C2) > Pi + (hProof. Let crj = sgnfo - r 0 ). Then \xi - x2\2 = | n - x 0 | 2 + \x2 - x0\2 - 2 ( i ! - x 0 ) • (x 2 - x 0 ) , In - r 2 | 2 = | n - r0\2 + |r 2 - r 0 | 2 - 2(rx - r 0 )(r 2 - r 0 ) , |*i - Z2I2 - In - r 2 | 2 = | i ! - xo| 2 - | n - r0\2 + |x 2 - xo| 2 - |r 2 - r 0 | 2 + 2a 1 a 2 (|r 1 - r 0 ||r 2 - rQ\ - |x x - x 0 ||x 2 - x 0 |) + 2-^-Q2>^(/31+/32)>/31+^2. Pl+ P2
□
Using Lemma 3.4.2 we can deal with the case /? < 10e that was left open in Lemma 3.4.1. As suggested by the case where C\ and C2 are tangent, we will show, roughly speaking, that any circle C G C^ 1 n C ^ 2 has to intersect the arc of minimal length on C\ that contains C\ n C2. Lemma 3.4.3. Suppose C2 G C%}. Then l # n C £ | < £ ^ ± ^ .
(3.44)
Proof. We may assume that T < 4t (otherwise C^ 1 n C£ 3 = 0). Let
Suppose C G Cft1 satisfies min(Z(x, 11, x2), Z(x, xi, - x 2 ) ) > Aojo , AQ being the constant in (3.43). In view of (3.40) we apply Lemma 3.4.2 with Co = Ci, C\ = C2, C2 — C, Pi = /3, fa = e, pi = T, and p2 = t to wit A(C,,C2)>£ + / 3 > e . In particular C ^Cft3. We conclude that any C G C^1 n Cft3 has to satisfy m i n ( Z ( x , x i , X 2 ) , / ( x , x i , - x 2 ) ) < Aojo ■
1? —> Lq Estimates for the Circular Maximal Function
51
In particular, the centers of all circles in C^1 n C%? are contained in a At x It A)7o rectangle centered at xi and thus
as claimed.
D
The following result is the aforementioned two-circle lemma. L e m m a 3.4.4. Suppose C2 G C^I. Then |C£nC5'l Lq Estimates for the Circular Maximal Function
59
11. J. Peral, V estimates for the wave equation, J. Funct. Anal. 36 (1980) 114-145. 12. W. Schlag. A generalization of Bourgain's circular maximal theorem, preprint 1995, to appear in J. Amer. Math. Soc. 13. W. Schlag and C. Sogge, Local smoothing estimates related to the circular maxi mal theorem, preprint 1996, to appear in Math. Res. Lett. 14. A. Seeger, C. Sogge and E. Stein, Regularity properties of Fourier integral oper ators, Ann. Math. 134 (1991) 231-251. 15. C. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991) 349-376. 16. C. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathe matics #105 (Cambridge Univ. Press, 1993). 17. E. Stein, Singular Integrals and Differentiability Properties of Functions (Prince ton Univ. Press, 1970). 18. E. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976) 2174-2175. 19. E. Stein, Harmonic Analysis, Princeton Math. Series 43 (Princeton Univ. Press, 1993). 20. E. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978) 1239-1295. 21. M. Talagrand, Sur la measure de la projection d'un compact et certaines families de cercles, Bull. Sci. Math. 104 (1980) 225-231. 22. T. Wolff, A Kakeya type problem for circles, preprint 1996, submitted to Pac. J. Math. 23. W. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics #120 (Springer-Verlag, 1989).
61
T H R E E R E G U L A R I T Y RESULTS I N H A R M O N I C ANALYSIS
Terry Tao Department of Mathematics, University of California at Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095, USA E-mail:
[email protected] In this dissertation we prove certain regularity properties of three unrelated families of operators arising from separate problems in harmonic analysis. The first result concerns the classical Bochner-Riesz operators Ss on Euclidean spaces R n (as well as more general Riesz means on manifolds). By the work of C. Fefferman, P. Tomas, E. Stein and M. Christ, one can obtain regularity results on these operators from l? restriction theory. We encompass these re sults by using the restriction theorem to prove an optimal weak-type estimate for the index S = (n — l)/[2(n -I-1)], which is the sharpest possible result one can obtain from L 2 restriction theory alone. The second result addresses the question of the pointwise convergence of various wavelet sampling methods. In applications one often samples a func tion / = ^2. . ajiici>j,k by discarding all but the largest wavelet coefficients, leaving a reconstructed function of the form ^ . . . a^V^fe- We show that under general conditions the sampled function converges pointwise almost ev erywhere to the original function as A —> 0. This is achieved by approximating the sampling operator by a linear sampling method whose convergence was established by Kelly, Kon and Raphael. Our final result extends recent work by M. Christ, J. L. Rubio de FVancia and A. Seeger, on weak (1,1) estimates for rough operators in Euclidean spaces, to more general homogeneous groups. This is still a work in progress by the author, and the results presented here are somewhat partial in nature. We show that a homogeneous singular integral convolution operator is of weak-type (1,1) if it is bounded on L2 and the kernel is L log L on the sphere.
1. Genera] Notation For further details on some of the terms used below, the reader is referred to Stein, 30 Meyer20 or Folland and Stein. 16
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We shall use the symbol C to denote various large positive constants depending only on the inessential variables, and e to denote various small con stants. In this section the inessential variables are the dimension n, the mani fold M, and the pseudo-differential operator P. In Sec. 2 the only inessential variable is the wavelet rp. In Sec. 3 the only inessential variable is the underly ing homogeneous group H. We shall write x = 0{y) or x < y for the statement that |x| < Cy, and write x ~ y if x < y and y < x. If x is a variable of integration, we use dx to denote Lebesgue measure and d#(x) to denote counting measure. When E is a set we use \E\ to denote the Lebesgue measure of E, and #E to denote the cardinality of E. On a Euclidean space R n we use ( , ) to denote the inner product, and define the Fourier transform by /(£) = / e - 2 , r ' ( x , ^ / ( x ) dx. If <j> is a function on R n , we define the Fourier multiplier operator 4>{D) by
#B)/(0 = *(0/(0More generally, we shall use the functional calculus to define operators <j>{P) whenever P is a suitable operator. For example, if P is self-adjoint and has the spectral resolution P = ^ j l j ^j£j, where the Xj are a discrete set of eigenvalues and e, are a mutually orthogonal spanning set of projections, then we can formally define <j>(P) by <j>{P) = Z ) j l i < H^j) e .r Let T be a linear operator. We use T[x, y] to denote the (distributional) kernel of T; thus
Tf(x) = JT[x,y\f(y)dy. If T[x, y] is supported on the set {(x, y) : d(x, y) < CK) where d is an ap propriate metric and R is a positive real, then we say that T is local at the scale of R. Finally, if A and A+ are linear operators, we say that A+ majorizes A if \A[x,y]\ < CA+[x,y] for all x, y, or equivalently that A+ is positivity preserving and \Af\ < CA+\f\ for all functions / . The following notation will be described for balls in a quasi-metric space, but will also be applied to Euclidean dyadic cubes. Suppose that H is a space with a quasi-distance d(x, y), such that the balls B(x, r) = {y: d(x,y) < r} satisfy the usual Vitali-type doubling and covering properties. li I = B(x,r) is a ball (or cube) and c > 0, we write r(I) for the radius (or sidelength) of / , xj for the center of / , cl for the dilate B(x, cr), and I& for an annulus of the form Cl — C~lI. If / is dyadic in the sense that r is a power of 2, and
Three Regularity Results in Harmonic Analysis
63
both / and i appear in an expression, we shall use the convention that i is the integer such that 2' = r{I) unless i is otherwise defined. By a wavelet we shall mean a 0-regular orthogonal wavelet tp on R, in the sense of Meyer.20 In other words, we require ip to be a function on R that is bounded and rapidly decreasing, satisfies the moment condition J i/;(x)dx = 0, and is such that the family {ipj,k} = {2j/2ip(2j • -k)} for j , k 6 Z form an orthonormal basis of L 2 (R). Note that no regularity assumptions are assumed on V; hi particular, our definition of a wavelet will include the Haar wavelet i> = X[0,l/2) - X [ l / 2 , 1 ) -
A homogeneous group shall be a canonical Euclidean space R n equipped with a group multiplication xy = x + y + P(x, y) and inverse x~l = —x + Q(x), where P and Q are polynomials consisting only of second-order and higher terms, and a family of dilations 5o(x1,...,xn)
=
(6a*x1,...,6a'>x)n)
for some constants 0 < a i < • • • < a n , such that i K+ J O I is a group automorphism for each 5 > 0. It can be shown that Lebesgue measure dx is both left- and right-invariant, and \S o E\ = 6N\E\ for all measurable sets E, where iV = QI + • • • + a n is the homogeneous dimension of H. We give H a quasinorm p by declaring p(x) = 1 on the Euclidean unit sphere S = {x: \x\ = 1}, and p(6 o x) = 6p{x) for all 8 > 0, x G H. The function d(x,y) = p(x-1y) thus becomes a quasi-distance. Let / be a function on some ball B(x, r) in a homogeneous group. We say that / is a normalized bump function adapted to I if f(xr °y) = 2. It is conjectured that, for 1 < p < 2n/(n + 1), these operators are of strong type (p,p) precisely when S > S(p) — n{\/p - 1/2) - 1/2. Progress on this conjecture can be found in Fefferman,12 Carleson-Sjolin,4 Bourgain, 2 Wolff;39 it remains open for n > 2 and p close to 2n/(n + 1 ) . A natural strengthening of the above conjecture is the claim that 5*W is of weak type (p,p) whenever* 1 < p < 2 n / ( n + 1 ) . This claim has been verified "The necessity of the condition on p was shown in Fefferman.13
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T. Tao
when n = 2 (Seeger 23 ), or when the operator is restricted to radial functions (Chanillo and Muckenhoupt 5 ). In Christ, 7,8 the claim is proved for p < po, where po is such that there is a (po,2) restriction theorem for the sphere in R n . The well-known restriction theorem of Tomas and Stein states that such a restriction theorem holds precisely when po < 2(n + l ) / ( n + 3). Thus the conjecture is known when b 1 < p < 2(n + l ) / ( n + 3). Our first result is that the conjecture also holds at the endpoint p = 2(n 4l ) / ( n + 3). Theorem 2.1. Suppose there is a (p, 2) restriction theorem for the sphere in R n . Then the operator Ss^ is of weak type (p,p). In fact, under the restriction theorem hypothesis we have the stronger es timate / |^)/(x)|2dx 2, and assume that 1 < p < 2 is such that a (p, 2) restriction theorem holds for the sphere. Write Ss = ms(D), where m'(£) = (1 - |£|2)$. and S = 8(p) = n ( l / p - 1/2) - 1/2. We have to show that ms(D) is of weak type (p,p). Since Mp is of weak type (p,p), it suffices by Tchebyshev's inequality to prove (1). By linearity we may assume that a = C~l for some large C to be determined later. Fix / , and apply the Calderon-Zygmund decomposition at height C to | / | p . This allows us to write / = g + J2] f>h where ||g||oo ^ 1) the bj are supported on disjoint dyadic cubes / and satisfy ||6/|| p ~ \I\l^p, and the I are such that £ / |J| ;$ 11/11?. Note that Mpf(x) > 1 whenever x e (J/ CI.
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T. Too
Since ms is bounded, the contribution of g to (1) is acceptable. In fact, as m is also compactly supported, we may similarly dispose of the contribution of the small cubes. 0 Specifically, let g be g + J2i)
