Calder on-Zygmund theory Updated May 23, 2020 Plan 2 Outline: - - PowerPoint PPT Presentation

Calder´

• n-Zygmund theory

Updated May 23, 2020

Plan

2

Outline: Statement and motivation Proof via Marcinkiewicz and duality Applications to Hilbert and inverse-Fourier transform General Calder´

• n-Zygmund kernels

Motivations

3

Q: What happens with Riesz transform Tαfpxq :“ ż 1 |x ´ y|α fpyqdy when α “ d? Or with Hilbert transform? A: Integral not defined even for nice f due to singularity at x “ y, but could truncate to |x ´ y| ě ǫ, perhaps. Singularity as |x| Ñ 8 bad too; kernel Kpxq :“ 1 |x|d 1|x|ěǫ

• beys K P L1,w, but Schur’s test requires Lr,w with r ą 1

Calder´

• n-Zygmund theorem

4

Theorem (Calder´

• n-Zygmund)

Consider the measure space pRd, LpRdq, λq for d ě 1. For all A, B ą 0, all M ą 1 and all p P p1, 8q there is Cp P p0, 8q such that for all measurable kernels K: Rd Ñ R satisfying K P L2 with Fourier transform p K obeying }p K}8 ď A and sup

zPRdt0u

ż

|x|ąM|z|

ˇ ˇKpx ´ zq ´ Kpxq ˇ ˇ dx ď B, the convolution operator TKf :“ K ‹ f is well defined by the integral expression for all f P L1 and extends continuously to a map Lp Ñ Lp for each p P p1, 8q with @p P p1, 8q: }TK}LpÑLp ď Cp

Remarks

5

1st condition ensures K is locally integrable and K ‹ f meaningful for f P L1 (by Young convolution inequality) 2nd condition often stated as K P C1pRd t0uq with @x P Rd t0u: ˇ ˇ∇Kpxq ˇ ˇ ď ˜ B |x|d`1 which (along with 1st condition) gives |Kpxq| ď ˜ B{d |x|d and so we cannot hope for more than K P L1,w (and so Schur’s test is still out). Upshot: Trading local regularity against integrability

Strategy of proof

6

1st condition implies TK is strong type p2, 2q with 2nd condition this implies TK is weak type p1, 1q. This is the key novelty; requires so called Calderon-Zygmund decomposition of Rd into sets where f is bounded and sets where f has bounded integral Marcinkiewicz interpolation gives TK is strong type pp, pq for all p P p1, 2s Duality: true also for p P r2, 8q

Improved Young convolution inequality

7

Recall: By Young convolution inequality f P L2 and g P L1 implies integral f ‹ g converges absolutely a.e. and }f ‹ g}2 ď }f}2}g}1 Need a slight improvement: Lemma @f P L1 @g P L2 : }f ‹ g}2 ď }f}2 }p g}8 Proof: Let f P L1 and g P L2. Fourier transform isometry so } y f ‹ g}2 ď }f}1}g}2 Hence g ÞÑ y f ‹ g continuous. If g P L1, then y f ‹ g “ p fp g so true for g P L2 as well. Hence }f ‹ g}2 “ } y f ‹ g}2 “ }p fp g}2 ď }p f}2}p g}8 “ }f}2}p g}8

Strong type p2, 2q

8

Corollary The operator TK is strong type p2, 2q with }TK}L2ÑL2 ď A Proof: For f P L1, TKf well defined via K ‹ f and obeys }TKf}2 ď A}f}2 by above lemma. So TK extends to L2 with }TK}L2ÑL2 ď A

Calder´

• n-Zygmund decomposition

9

Dyadic cube is any cube of the form 2nx ` r0, 2nqd for x P Zd and n P Z. Lemma (Calder´

• n-Zygmund decomposition)

Let f P L1 and t ą 0. Then there exist disjoint dyadic cubes tQiuiPI such that @i P I: tλpQiq ă ż

Qi

|f|dλ ď 2dtλpQiq and |f| ď t λ-a.e. on Rd ď

iPI

Qi

Proof of Calder´

• n-Zygmund decomposition

10

Call a dyadic cube Q good if 1 λpQq ż

Q

|f|dλ ď t and call it bad otherwise. For n P Z such that ş |f|dλ ď t2n, all dyadic cubes of side-length 2n good. Let tQiuiPI enumerate the set of all bad dyadic cubes Q such that the (unique) dyadic cube Q1 containing Q and having side length twice as that of Q is good. Then tλpQq ă ż

Q

|f|dλ ď ż

Q1 |f|dλ ď tλpQ1q “ 2dtλpQq

because Q is bad and Q1 is good. If x lies only in good cubes, Lebesgue differentiation shows |fpxq| ď t a.e. (Need a version for dyadic cubes; proved when discussed martingale convergence.)

Weak type p1, 1q

11

Proposition TK is weak type p1, 1q. Explicitly, Dc P p0, 8q @f P L1 @t ą 0: λ ` |TKf| ą t ˘ ď c t}f}1 where c depends only d and the constants A and B

Decomposition of f

12

Pick f P L1 and t ą 0 and let tQiuiPI be as above. Set F :“ Rd ď

iPI

Qi define g: Rd Ñ R by gpxq :“ #

1 λpQiq

ş

Qi fdλ

if x P Qi for some i P I fpxq if x P F and abbreviate hpxq :“ fpxq ´ gpxq Note that h “ 0 on F ^ @i P I: ż

Qi

hdλ “ 0 Union bound + additivity: λ ` |TKf| ą t ˘ ď λ ` |TKg| ą t{2 ˘ ` λ ` |TKh| ą t{2 ˘ Now estimate each term separately . . .

Tails of TKg

13

Will use that TK maps L2 Ñ L2 with norm ď A (proved above). Need to estimate }g}2

2 “

ż

F

g2dλ ` ÿ

iPI

ż

Qi

g2dλ ď ż

F

t|f|dλ ` ÿ

iPI

´ 1 λpQiq ż

Qi

fdλ ¯2 λpQiq ď ż

F

t|f|dλ ` ÿ

iPI

p2dtq2λpQiq ď t ż

F

|f|dλ ` 4dt ÿ

iPI

ż

Qi

|f|dλ “ t ż

F

|f|dλ ` 4dt ż

Fc |f|dλ “ p4d ` 1qt}f}1

Hence λ ` |TKg| ą t{2 ˘ ď 4 t2 }TKg}2

2 ď 4A2

t2 }g}2

2 ď 4A2p4d ` 1q

t }f}1

Tails of TKh

14

Consider hi :“ h1Qi. Let yi :“ the center of Qi. As ş

Qi hdλ “ 0,

TKhipxq “ ż

Qi

Kpx ´ yqhipyqdy “ ż

Qi

` Kpx ´ yq ´ Kpx ´ yiq ˘ hipyqdy Let Q1

i :“ the cube of M

? d-times the side length of Qi centered at yi. By Tonelli and 2nd condition: ż

RdQ1

i

|TKhi|dλ ď ż

Qi

´ż

RdQ1

i

ˇ ˇKpx ´ yi ` y ´ yiq ´ Kpx ´ yiq ˇ ˇdx ¯ˇ ˇhipyq ˇ ˇdy ď B ż

Qi

|h|dλ ď 2B ż

Qi

|f|dλ which uses |x ´ yi| ą M|y ´ yi| for all x R Q1

i and all y P Qi. Then

. . .

Tails of TKh continued ...

15

. . . abbreviating F1 :“ Rd Ť

iě1 Q1 i we thus get

ż

F1

ˇ ˇTKh| dλ ď 2B}f}1 On the other hand, λpRd F1q ď ÿ

iPI

λpQ1

iq “ pM

? dqd ÿ

iPI

λpQiq ď pM ? dqd t ÿ

iPI

ż

Qi

|f|dλ ď pM ? dqd t }f}1 and so λ ` |TKh| ą t{2 ˘ ď λpRd F1q ` 2 t ż

F1

ˇ ˇTKh| dλ ď pM ? dqd ` 4B t }f}1 So claim holds with c :“ 4A2p4d ` 1q ` pM ? dqd ` 4B

Proof of Calder´

• n-Zygmund theorem

16

Marcinkiwicz: TK strong type pp, pq for p P p1, 2s. Now let q P p2, 8q and let p be such that p´1 ` q´1 “ 1. Then duality between Lp and Lq gives @f P L1 X Lp @g P Lq : ˇ ˇ ˇ ż gpK ‹ fq dλ ˇ ˇ ˇ ď }TK}LpÑLp}f}p}g}q For f P L1 integral K ‹ f converges absolutely. So by Fubini-Tonelli: @f P L1 X Lp @g P Lq X L1 : ˇ ˇ ˇ ż pTKgqf dλ ˇ ˇ ˇ ď }TK}LpÑLp}f}p}g}q Density of Lp X L1 in Lp gives @g P Lq X L1 : }TKg}q ď }TK}LpÑLp}g}q. which implies that TK extends continuously to a map Lq Ñ Lq with }T}LqÑLq ď }TK}LpÑLp (Equality holds by duality.)

Application to Hilbert transform

17

Recall: Hf defined as the ǫ Ó 0 limit of convolution-type

• perator Hǫf :“ Kǫ ‹ f where

Kǫpxq :“ 1 πx 1pǫ,1{ǫqp|x|q Convergence pointwise for f P C1pRq X L1 and in L2 for f P L2 Theorem (Hilbert transform in Lp) We have @p P p1, 8q: sup

0ăǫă1

}Hǫ}LpÑLp ă 8. In particular, for all p P p1, 8q, there exists a continuous linear

• perator H: Lp Ñ Lp such that

@f P Lp : Hǫf Ý Ñ

ǫÓ0 Hf in Lp.

Proof of Theorem

18

For strong type p2, 2q, use Fourier calculation to get p Kǫpzq “ ´2i π ż

ǫătă1{ǫ

sinp2πztq t dt Hence, A :“ sup0ăǫă1 }p Kǫ}8 ă 8. For weak type p1, 1q, compute ˇ ˇKǫpx ´ zq ´ Kǫpxq ˇ ˇ ď ˇ ˇ ˇ 1 x ´ z ´ 1 x ˇ ˇ ˇ ` 1 |x| ˇ ˇ1pǫ,1{ǫqp|x ´ z|q ´ 1pǫ,1{ǫqp|x|q ˇ ˇ ď 2|z| |x|2 ` 1 |x|1tp1{ǫ´|z|,1{ǫ`|z|qp|x|q ` 1 |x|1tpǫ´|z|,ǫ`|z|qp|x|q Need to integrate this over |x| ą 2|z|. First term easy. For the

• ther two terms we note that . . .

Proof of Theorem continued ...

19

. . . for any a, b ą 0 with maxt2b, a ´ bu ă a ` b, ż a`b

maxt2b,a´bu

dx x “ log ´ a ` b maxt2b, a ´ bu ¯ Examining a ´ b ă 2b and a ´ b ą 2b separately, RHS ď logp2q. Now use this with a :“ ǫ, 1{ǫ and b “ |z| to get B :“ sup

0ăǫă1

sup

zPRt0u

ż

|x|ą2|z|

ˇ ˇKǫpx ´ zq ´ Kǫpxq ˇ ˇ dx ă 8 So tHǫu0ăǫă1 obey conditions of Calder´

• n-Zygmund theorem

with uniform A and B (and M :“ 2). So we get sup

0ăǫă1

}Hǫ}LpÑLp ă 8. It remains to address convergence Hǫf Ñ Hf . . .

Proof of Theorem continued ...

20

. . . which we already know in L2 by Fourier techniques. We will use interpolation for Lp-norms. Given p P p1, 8q, choose ˜ p P p1, pq when p ă 2 or ˜ p P pp, 8q when p ą 2. Then 1

p “ p1 ´ θq1 ˜ p ` θ 1 2 for some θ P p0, 1q and so

@f P L˜

p X L2 :

}Hǫf ´ Hδf}p ď }Hǫf ´ Hδf}θ

2 }Hǫf ´ Hδf}1´θ ˜ p

. Now }Hǫf ´ Hδf}2 Ñ 0 as ǫ, δ Ó 0 by the claim in L2 while }Hǫf ´ Hδf}˜

p ď

´ 2 sup

0ăǫ1ă1

}Hǫ}L˜

pÑL˜ p

¯ }f}˜

p.

Completeness of Lp shows Hǫf Ñ Hf for each f P L˜

p X L2. As

L˜

p X L2 dense in Lp, true for all f P Lp.

Uniform convergence?

21

Q: Is Hǫf Ñ Hf uniform in f P Lp with }f}p ď 1? A: Not in L2 (and by duality in interpolation, not in Lp) because }Hǫ ´ H}L2ÑL2 “ }p Kǫ ´ p K}8 and RHS does not tend to zero because Kǫ is continuous and p Kpzq :“ p´iqsgnpzq is not.

Strong vs norm convergence

22

Definition (Strong and norm convergence of operators) A sequence tTnuně1 of linear operators on a normed linear space V is said to converge strongly to a linear operator T if @f P V : lim

nÑ8 }Tnf ´ Tf} “ 0

The sequence tTnuně1 converges to T in (operator) norm if lim

nÑ8 }Tn ´ T} “ 0

So, on Lp with p P p1, 8q, we get Hǫ Ñ H strongly but not in

• perator norm.

Cotlar’s approach

23

Cotlar’s identity: pHfq2 “ f 2 ` H ` fpHfq ˘ By induction: For n ě 1 and p :“ 2n, }Hf}2p

2p ď p}f}2p 2p ` p

› ›HpfpHfqq}p

p

ď p}f}2p

2p ` pp}H}LpÑLpqp}f}p 2p}Hf}p 2p

Proves }H}LpÑLp ă 8 for p P t2n : n ě 1u. Interpolation + duality gives this for all p P p1, 8q.

Partial Fourier inversions

24

For f P L1, define Tnfpxq :“ ż n

´n

p fpkqe´2πik¨xdk, where p f :“ Fourier transform of f. Know that, if p f P L1, then Tnf Ñ f pointwise. Q: Convergence in Lp? Theorem Let p P p1, 8q. Then, for each n ě 1, the operator Tn extends continuously to a map Lp Ñ Lp and @f P Lp : Tnf Ý Ñ

nÑ8 f in Lp

Proof: homework

A.e. convergence

25

Lp convergence gives a.e. convergence along a subsequence. Need for subsequences removed by L. Carleson (1966) for L2 for L2 and by R. Hunt (1968) for Lp (1 ă p ă 8). Key idea: Carleson operator T‹fpxq :“ sup

ně1

ˇ ˇ ˇ ż n

´n

p fpkqe´2πik¨xdk ˇ ˇ ˇ, is weak type p2, 2q. Hard proof (ą 100 pages). M. Lacey and C. Thiele in “A proof

• f boundedness of the Carleson operator” (Math. Res. Lett. 7

(2000), no. 4, 361–370) give a proof in under 20 pages. No a.e. convergence for L1 functions (A.N. Kolmogorov’s counterexample)

Calder´

• n-Zygmund theory, general kernels

26

Definition (Calder´

• n-Zygmund type)

Given A, B ą 0 and M ą 1, a linear operator T: CcpRdq Ñ L0 we say that T is Calder´

• n-Zygmund type with parameters pA, B, Mq if

@f P CcpRdq: }Tf}2 ď A}f}2 and there is a measurable kernel K: Rd ˆ Rd Ñ R such that sup

yPRd

sup

zPRdt0u

ż

|x´y|ąM|z|

ˇ ˇKpx, y ` zq ´ Kpx, yq ˇ ˇ dx ď B for which T admits the integral representation @f P CcpRdq: Tfp¨q “ ż Kp¨, yqfpyqdy λ-a.e. with integral absolutely convergent λ-a.e.

Strong type p2, 2q

27

Note: Assuming strong type p2, 2q! Sufficient conditions exist: Lemma Let pX, F, µq and pY, G, νq be σ-finite measure spaces and let K: X ˆ Y Ñ R be F b G-measurable and such that K P L2pµ b νq. Then for each f P L2, the integral in Tfpxq :“ ż Kpx, yqfpyqνpdyq converges absolutely for ν-a.e. x P X and defines a continuous linear

• perator L2pνq Ñ L2pµq. Moreover,

}T}L2pνqÑL2pµq ď }K}L2pµbνq These are usually too weak to be used here. Strong type p2, 2q property usually verified by “Hilbert space” techniques.

Main theorem

28

Theorem (Calder´

• n-Zygmund, general form)

For all A, B ą 0, M ą 1, d ě 1 and p P r1, 2s there is Cp P p0, 8q such that for every Calder´

• n-Zygmund-type operator T with

parameters pA, B, Mq, we have: (1) T is weak type p1, 1q with (its extension to L1 satisfying) @t ą 0 @f P L1 : λ ` |Tf| ą t ˘ ď C1 t }f}1 (2) For each p P p1, 2s, T is strong type pp, pq with (its extension to Lp satisfying) @f P Lp : }T}p ď Cp}f}p If K‹px, yq :“ Kpy, xq is also C.Z.-type with the same A, B, M, then T is also strong type pp, pq for every p P r2, 8q with Cp :“ C

p p´1

Proof: main changes

29

The proof for p P p1, 2s is taken nearly verbatim. Duality argument requires some work. First some functional analysis: Lemma Let V be a normed linear space and let T: DompTq Ñ V be a linear

• perator on V with dense linear DompTq. For each φ P V‹,

@f P DompTq: pT‹φqpfq :“ φpTfq defines a linear functional T‹φ on DompTq. The map φ ÞÑ T‹φ is linear and so T‹ is a linear operator called the adjoint of T. If T is bounded, then T‹φ P V‹ and T‹ extends to a continuous linear

• perator T‹ : V‹ Ñ V‹ with

}T‹} ď }T}. (Equality holds by the Hahn-Banach theorem.)

Proof of Lemma

30

Linearity of T‹ clear. For T bounded, @f P DompTq @φ P V‹ : ˇ ˇpT‹φqpfq ˇ ˇ ď }φ} }Tf} ď }T} }φ} }f} As DompTq is dense in V, T‹φ extends continuously to V with }T‹φ} ď }T}}φ} Hence }T‹} ď }T}.

Adjoint of integral operators

31

Lemma Let p, q P p1, 8q be such that p´1 ` q´1 “ 1 and let T: CcpRq Ñ L0 be a linear operator such that @f P CcpRdq: Tfp¨q “ ż Kp¨, yqfpyqdy λ-a.e. with the integral convergent λ-a.e. If T is continuous as a map Lp Ñ Lp, its adjoint T‹ admits the integral representation @f P CcpRdq: T‹fp¨q “ ż Kpy, ¨qfpyqdy λ-a.e. where the integral converges λ-a.e.

Proof of Lemma

32

For f, g P CcpRdq, Fubini-Tonelli gives ż gpTfqdλ “ ż gpxq ´ż

|x´y|ąǫ

Kpx, yqfpyqdy ¯ dx “ ż fpyq ´ż Kpx, yqgpxqdx ¯ dy “ ż fpr Tgqdλ where r Tg :“ ş Kpy, ¨qfpyqdy converges absolutely. Riesz representation: φgpTfq “ φr

Tgpfq

Using that g ÞÑ φg is bijective isometry of pLpq‹ Ñ Lq, we now identify T‹φg with r Tg.

Proof of Calder´

• n-Zygmund theorem, p P r2, 8q

33

Pick p P p2, 8q and let q be H¨

• lder dual. Let r

T be defined using K‹-kernel (which is C.Z.-type). Then Lemma says r T‹ “ T (1) and so }T}LpÑLp ď }r T}LqÑLq ď Cq (2)