多項式調變不變奇異積分算子

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國立臺灣大學理學院數學系暨研究所碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

多項式調變不變奇異積分算子 Polynomial Modulation Invariant

Singular Integral Operator

徐永昌

Yung-Chang Hsu

指導教授：沈俊嚴副教授

Advisor: Chun-Yen Shen, Associate Professor

中華民國110年2月  February 2021

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摘要

本文針對多項式卡爾松算子高維推廣在勒貝格空間下的有界性作深入探討。相比於Victor Lie與Pavel Zorin-Kranich之前的工作，該文章的主要貢獻包含：以具體的構造法來確認細節論證、用稀疏算子的語言來重新詮釋部分證明、及提供一個具教學啟發性的完整說明。

關鍵詞：時頻分析、多重解析度分析、CZ算子、稀疏壓制、TT-TT方法

誌謝

作者感謝指導教授開明的態度以及十足的耐心，促成這漫長的計畫以及深

度的細部探討。其次作者感謝與多位同事日常討論所激發的靈感，造就了諸多

論證的精進。最後，作者想感謝家人以及身邊每位關心作者的人。

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Polynomial Modulation Invariant Singular Integral Operators

Yung-Chang Hsu Feb. 2021

Abstract

We deeply study the L^pboundedness of the generalization of Polyno- mial Carleson Operator. Our main contributions, comparing to previous works done by Victor Lie and by Pavel Zorin-Kranich, are to verify details with explicit constructions, modify some part with language of Sparse Dominance, and provide a heuristic interpretation about the whole treatment in general.

Keywords—Time-Frequency Analysis, Multi-Resolution Analysis, CZO, Sparse Domi- nance, TT* method

1 Introduction

There are three major themes in Harmonic Analysis that ordinary tools in Real Analysis are weak against:







Singular ⇒ Singular Integral Operator

Maximal ⇒ Hardy-Littlewood Maximal Operator Oscillatory ⇒ Fourier Integral Operator.

Still, mathematicians have developed tools for individual class of operators and have gained fruitful understanding. Before becoming overly optimistic, however, what if there is an instance where the three themes combine together?

Definition 1.0.1 (Carleson Operator).

Cf (·) := sup

N ∈R

p.v.

ˆ e^{iN y}

· − yf (y)dy .

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Indeed, we see that there are:

• Singularity in the integral kernel _·−y¹ .

• Pointwise maximal in the evaluation.

• Oscillation within the integral.

Naturally, we can not expect the tools designed for one particular theme to be effective against such operator. Maybe, we just need to combine all the tools in a smart ways. Additionally, we better do so in a way that separate different features from different themes so that each individual tools can shine. In hindsight, the missing glue to stick all the tools together is Time- Frequency Analysis. While, the participation of sparse dominance is a pleasant surprise.

Of course, this operator is not something mathematicians conjure up just for fun. To convince the reader that such type of operators arises naturally, we first introduce some notions.

1.1 Basic Notions

As a preparation for stating the main result, we introduce some definitions and notations. Throughout this thesis, we only work under Euclidean setting (R^D).

Definition 1.1.1.

Qd:= {q ∈ R[x¹][x2] · · · [xD] | degq ≤ d}

Definition 1.1.2 (Standard Kernel).

Given K : R^D× R^D→ C, we say K is a Standard Kernel if given x, y ∈ R^D, we have ”Size Control”:

|K(x, y)| . kx − yk^−D.

Furthermore, there’s τ ∈ (0, 1] such that for ∆ ∈ R^D satisfying _kx−yk^k∆k ≤ ¹₂, we also have ”τ -H¨older Type Control”:

|K(x + ·, y)

∆

0| + |K(x, y + ·)

∆

0| . (k∆k/kx − yk)^τ kx − yk^D . Definition 1.1.3 (Calderon-Zygmund Operator).

Given T ∈ BL(L², L²), we say T is a Calderon-Zygmund Operator (CZO) if it’s associated to a standard kernel K in the following sense:

∀f, g ∈ C_c^∞, suppf ∩ suppg = ∅ ⇒ hT f, gi = ˆ

K(x, y)f (y)g(x)dxdy.

Remark. Kernel determines a CZO up to a difference of Multiplication Opera- tor. That is: Given T, S ∈ BL(L², L²) be a pair of CZOs, if T, S are associated to the same kernel, then

∃m ∈ L^∞ s.t. ∀f ∈ L², T f − Sf = mf.

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For the rest of the thesis, we fix T ∈ BL(L², L²) a CZO, denote the corresponding kenerl as K(·, ·), and use f ∈ C_c^∞to denote a generic function. Now, we introduce some related operators.

Definition 1.1.4 (Singular Integral Operator).

If the kernel satisfies additional regularity condition:

∀⁰x ∈ R^D, lim

→0⁺

ˆ

≤kx−yk≤1

K(x, y)dy exists,

the following limit:

lim

→0⁺

ˆ

≤k·−yk

K(·, y)f (y)dy

actually defines a CZO associated to K. We call this particular type of CZO Singular Integral Operator.

Definition 1.1.5 (Maximal Truncated CZO).

T_∗f (·) := sup

r<R

ˆ

r≤k·−yk<R

K(·, y)f (y)dy .

Definition 1.1.6 (Maximal Operator).

Mrf (·) := sup

B3·

|f |B,r

where B denotes a cube and |f |B,r := ffl

B|f |^rdµ1/r

with r ∈ [1, ∞) and µ the Lebesgue measure. Notice that Hardy-Littlewood Maximal Operator is essentially the case when r = 1. For convenience, we write:

M f := M1f and |f |_B := |f |_B,1. Definition 1.1.7 (Polynomial Modulation Invariant CZO).

Cdf (·) := sup

q∈Qd

T (e^iqf )(·)

Definition 1.1.8 (Maximal Truncated Polynomial Modulation Invariant CZO).

Cd∗f (·) := sup

q∈Qd

T_∗(e^iqf )(·)

Observation. Due to a version of Cotlar’s Inequality([Duo+01]Lemma 5.15), we always have:

T_∗f . M T f + M f, and thus,

Cd∗f . M C^df + M f.

As a result, boundedness of Cd implies boundedness of Cd∗.

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1.2 Motivation

We provide some instances where considering such type of operators are relevant.

• Pointwise a.e. Convergence of Fourier Series: In 1915, Luzin conjectured that the Foruier series of a L²function converges almost ev- erywhere to the function itself. The result is proved fifty years afterward.

Theorem 1.2.1 (Carleson’s Theorem).

Qualitative statement:(Lennart Carleson, 1966 [Car66])

The Fourier Series of L² function converge a.e. to itself.

Quantitative statement:(Charles Fefferman, 1973 [Fef73]) T be Hilbert Transform on T, kC¹f k_L1(T). kf kL²(T).

The original proof was quite complicated. It was not until 1973 that Fefferman gave a much elegant proof on the quantitative equivalence based on Stein Maximal Principle and ideas of Time-Frequency Analysis.

• Constant Coefficient PDE: We provide the most elementary case:

Heat equation to illustrate the idea.

(u_t(x, t) − ∆_xu(x, t) = f (x, t), t > 0 u(x, 0) = u0(x),

Due to the linearity of the equation, we reduce to solve the following two sets of equation:











(u_t(x, t) − ∆_xu(x, t) = 0, t > 0

u(x, 0) = u0(x) homogeneous

(ut(x, t) − ∆xu(x, t) = f (x, t), t > 0

u(x, 0) = 0 non-homogeneous.

Suppose we have understood how the regularity of the initial data u0

affects the regularity of the solution u of the homogeneous equation.

We now proceed to investigate how the non-homogeneous term f affects the regularity of the solution u in the sense of Sobolev space language. To do so, we first assume the following stronger condition:

Given u(·, ·), f (·, ·) ∈ S(R^D× R) that vanishes for t ≤ with some > 0, ut(x, t) − ∆xu(x, t) = f (x, t)

Fourier

⇐⇒ 2πiτ + 4π²|ξ|²

u(ξ, τ ) = bb f (ξ, τ ) By defining m(ξ, τ ) := _{2πiτ +4π}^2πiτ2|ξ|² and setting

Ltf := F⁻¹(mF(f )) ,

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we expect Ltf to solve ut. Notice that m(λξ, λ²τ ) = m(ξ, τ ), thus by setting K := F⁻¹(m), we have:

K(λx, λ²t) = λ^−D−2K(x, t).

As we expand L_tf :

Ltf (·) := K ∗ f (·)

= ˆ

R+

ρ^D+2 ˆ

S^D

K(ρx, ρ²t)f (· − (ρx, ρ²t))J (x, t)d(x, t)dρ ρ

= ˆ

S^D

K(x, t)J (x, t) ˆ

R+

f (· − (ρx, ρ²t))dρ ρd(x, t), we reduce to control the following operator:

Definition 1.2.2 (Hilbert Transform Along Paraboloid).

H_(y,s)f (x, t) := p.v.

ˆ

R

f ((x, t) − (ρy, ρ²s))dρ ρ

Denoting Fourier on (˜x, t) := (x2, x3, · · · , xD, t)” as ˜F, we deduce:

F˜ H_(y,s)f (·, ˜ξ, τ ) = p.v.

ˆ

R

e^−2πi(^ρ²^{τ s+ρ ˜}^ξ·˜^y) ˜Ff (· − ρy₁, ˜ξ, τ )dρ ρ, which can be controlled by C2 with T be Hilbert Transform:

F˜ H_(y,s)f (·, ˜ξ, τ )

.C₂Ff (·, ˜˜ ξ, τ ).

If we have kC₂f k_L2 . kf kL², then using the tensor product structure of the product measure and Plancherel theorem, we have:

∵

F˜ H_(y,s)f (·, ˜ξ, τ ) _L₂ .

Ff (·, ˜˜ ξ, τ ) _L₂

∴

H_(y,s)f L²=

F˜ H_(y,s)f _L₂ .

Ff˜

_L₂ = kf k_L2.

This implies that kL_tf k_L2 . kf kL². (There is an analogous statement for

∆_xu.) As a result, we can use density argument to infer that:

∀f ∈ L²(R^D×R+), ∃u solving the equation s.t. ut, ∆xu ∈ L²(R^D×R+), which can be easily translated to Sobolev space language.

Remark. If D = 1, the linear term in the modulation vanishes. This case is covered by Stein and Wainger’s result in [SW01]

• Modulation Symmetries: An operator may possess certain symmetry.

One such instance is polynomial modulation symmetry. We expect that understanding Cd and Cd∗ paves the way to the understanding of some more complicated operators.

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– Explicit Polynomial Modulation Invariance: (Hard but have result on L^p→ L^p boundedness.)

q ∈ Qd=⇒

(Cd e^iqf

= Cd(f ) C_d∗ e^iqf

= C_d∗(f )

– Implicit Polynomial Modulation Symmetry: (No good result on the boundedness of the operator for n > 2)

H−→α(fj)ⁿ_j=1(·) : = p.v.

ˆ

R n

Y

j=1

fj(· − αjt)dt t

n

X

j=1

qj(· − αjt) = q(·) =⇒ H⁻^→_α e^iq^jfjⁿ

j=1= e^iqH⁻^→_α(fj)ⁿ_j=1 Indeed, inspired by Fefferman’s proof on the boundedness of C1, Thiele and Lacey came up with a much elegant argument using the same philosophy to prove the boundedness of H_(1,−1):

Theorem 1.2.3 (Christoph Thiele & Michael Lacey, 1997 [LT97]).

∀p, q, r ∈ (2, ∞) such that ¹_p+¹_q +¹_r = 1,

|hH(1,−1)(f, g), hi| .

p,q,r

kf kL^pkgkL^qkhkL^r.

Later on, they notice the similarity (similar modulation symmetry) between C1 and H_(1,−1)and use their method to prove:

Theorem 1.2.4 (Christoph Thiele & Michael Lacey, 2000 [LT00]).

kC1f kL^2,∞. kf k^L², where T is Hilbert Transform.

It is tempting to believe that there is an implicit correspondence:

C_df, C_d∗f ^⇐=_=⇒ H−→α(f_j)ⁿ_j=1.

However, there must be some missing links between the two scenarios.

To elaborate, we present some of the differences:

(=⇒) We need to find a way to convert the multilinear nature of the operator into products of linear structures. Additionally, we better extract the implicit modulation symmetry into the form of explicit modulation invariance.

(⇐=) The conversion of C1into H_(1,−1)-like operator, relies on the Fourier correspondence between linear modulation and translation.

There is no good notion for polynomial modulation.

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• Detection of the Singularity: It is an idea from one of my colleagues.

Let us compare ¹_· and _|·|¹ and its corresponding operators:

(Hf (·) := p.v´ ₁

·−yf (y)dy Xf (·) := p.v´ ₁

|·−y|f (y)dy.

Some easy verification shows that:

(kHf k_Lp. kf kL^p

kXf k_Lp6. kfkL^p

, ∀p ∈ (1, ∞).

As we put in modulation: Fixing Q ⊂ C^∞(R, R), we define:

Qf (·) := sup

φ∈Q

p.v.

ˆ 1

· − ye^iφ(y)f (y)dy ,

we see that the behavior of Q is morally governed by the two cases: H and X. That is, if Q is too large, we can expect the modulation recovers the absolute value that is:

|Xf | ≤ |Qf | and, thus, kQf k_Lp6. kfk_Lp, ∀p ∈ (1, ∞).

Otherwise, we have for example: Q := Q1 and T := H,

C₁f = Qf and, thus, kQf k_Lp. kf kL^p, ∀p ∈ (1, ∞).

The interesting part is to find the borderline between the two cases:

Definition 1.2.5 (Detection of Singularity).

Given T ∈ BL(L², L²) a CZO, we say Q ⊂ C^∞(R^D, R) detects the singularity at p ∈ (1, ∞) if the operator defined as:

Qf (·) := sup

φ∈Q

T e^iφf (·) is not bounded at p. That is, kQf k_Lp6. kfk_Lp.

In other words, Q1 does not detect the singularity of Hilbert transform. We think a non-trivial example of Q that detects the singularity at specific p would give us new light on the understanding of the singularity of an operator.

1.3 Main Result

Stein conjectured that Cd is bounded for suitable K(·, ·). In his joint work with Wainger [SW01], a restricted case (excluding linear modulation) is resolved through the technique of stationary phase formula and T T^∗-T^∗T arguments. While, Lie, after proving the weak(2, 2) bound of C2with T being Hilber transform on T, proved the Stein conjecture for the following case:

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Theorem 1.3.1 (Victor Lie, 2020 Annals of Mathematics [Lie20]).

T be Hilbert Transform on T, kCdf kL^p(T) .

p,d

kf kL^p(T), ∀p ∈ (1, ∞)

Inspired by the proof, Zorin-Kranich extended the result and resolved the full Stein conjecture:

Theorem 1.3.2 (Pavel Zorin-Kranich, 2019 [Zor19]).

For arbitrary D, T ,

kCd∗f kL^p .

T ,D,d,p

kf kL^p, ∀p ∈ (1, ∞).

Remark. The precise condition for Theorem1.3.2is actually weaker:

kT_∗f k_Lp .

p

kf k_Lp, ∀p ∈ (1, ∞).

That is, even if there is no C.Z.O associated to the kernel K(·, ·), the condition is still valid. Alternatively, it infers that polynomials with bouneded degree cannot detect the singularity of the kernel if T^∗ is bounded.

By previous observation, it’s tempting to think C_d a more fundamental ob- ject and try proving its boundedness first. Naturally, we would come up with our first guess:

Theorem 1.3.3.

If T is a Singular Integral Operator, we always have:

kCdf kL^p .

T ,D,d,p

kf kL^p, ∀p ∈ (1, ∞)

However, in hindsight, we actually treat Theorem1.3.3as a direct corollary of Theorem1.3.2. Notice that it’s quite different from the treatment in [Lie20].

The author proves Theorem1.3.3for T being Hilbert Transform directly. We will address what causes the difference in3.4.

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2 Mathematical Jigsaw Puzzle

In this section, we give a heuristic explanation about how we’ll use Time- Frequency Analysis to proceed with the proof of Theorem1.3.2.

2.1 Cut out the Pieces

The idea is to linearize Cd∗: Cd∗f (·)

ˆ

r_(·)≤k·−yk<R_(·)

K(·, y)e^iq^(·)^(y)f (y)dy =: ˜Cd∗f (·)

so that the time-frequency information of f (·) gets transferred to the operator itself. Since q_(·) is encoded with the sheet music–time-frequency portrait of f (·), Time-Frequency Analysis would be done on ˜Cd∗ instead of f .

Next, we break ˜Cd∗ into tiny pieces and treat them as mathematical jigsaw puzzles. Our goal is to fit those pieces into a ”bounded” box. To do so, we do the following decomposition:

• Scale(s ∈ Z): We break K(·, ·) according to scales so that each piece mimics the behavior of a wavelet. As a result, the s-scale piece of the operator extracts 2^s-resolution features only. In short, we have

K(x, y) ∼X

s∈Z

wavelets(x−y) ∧ T∗f (·) ∼ sup

s<s





s

X

s=s

wavelets



∗ f (·) .

• Temporal block(I ⊂ R^D): With a fixed scale, we decompose the piece to separate the support into different temporal position with block-size matching the scale.

• Spectral block(ω ⊂ Qd): Fixing scale and temporal position, we decompose the piece again so that q_(·) fall in distinct spectral position with block-size respecting some kind of Uncertainty Principle.

That is, a generic piece satisfies:

2^s∼ diameter of I ∼ diameter of ω⁻¹,

where s is the natural scaling of I × ω and is denoted by s_I×ω. In short, C˜d∗f (·) ∼X

piece_I×ωf (·), where

piece_I×ωf (·) ∼ wavelets_I×ω∗ e^iq^(·)f (·)χE_I×ω(·) with

EI×ω:= {x ∈ I | qx∈ ω ∧ rx≤ 2^s^I×ω ≤ Rx} .

Naturally, this comes with good properties. For instance, all the pieces have similar sizes in BL(L^p, L^p). However, we need finer estimation, and we do so by tracking the following attributes for each piece:

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• Scale : This corresponds to the resolution of features the operator detects/takes in.

• Tile position : This refers to the position of tile P := IP× ωP on the time-frequency phase plane.

• Density : This measures how large portion of IP gets sent through q_(·) to ωP within the acceptable scale range. That is, A(P ) := ^|E_|I^P^|

P|. As an immediate result,

kpiece_Pf kL^p. A(P )^1/pkf kL^p.

This provides us with some intuition. By classifying the pieces according to their density (i.e. A(P ) h 2⁻ⁿ), we just need to remember extracting the 2⁻ⁿ- factor from our arguments. Namely, we shall focus on Pⁿ:= {P | A(P ) h 2⁻ⁿ}.

(Details would be made precise in4.)

2.2 Find Good Configurations

Up till now, we’ve reduced the puzzle to Pⁿ sub-puzzle. To proceed, we need to know how well pieces can be packed together in BL(L^p, L^p). Naturally, a good starting point would be BL(L², L²). This way, we can use Orthogonality to help us organize our pieces. As expected, ∃ > 0, s.t. ∀P_j ∈ Pn

(hpiece_P₀f, piece_P₁f i = 0 ⇐= P0∩ P1= ∅ hpiece^∗_P₀f, piece^∗_P₁f i

. 2⁻ⁿ(1 + distanceP0,P1)⁻. (7.1.3) Alternatively, if P ⊂ Pⁿ cluster at a spot (ξ, η) ∈ R^D × Qd, the cluster cluster_Pf := X

P ∈P

piece_Pf will extract distinct 2^s^P-resolution features of f

near (ξ, η). Therefore, provided that

({sP}_{P ∈P}= {s ∈ Z | s ≤ s ≤ s}

∀P ∈ P, distanceP,(ξ,η) 1 , we have qx∼ η as long as x ∈ [

P ∈P

EP is around ξ, and Multi-Resolution Anal- ysis yields

|cluster_Pf | ∼





s

X

s=s

wavelets



∗ (e^iηf )

χ₂⁻ⁿ_{-dense set}

around ξ

. T∗(e^iηf )χ₂⁻ⁿ_{-dense set}

around ξ

.

(8.2.3)

Moreover, by viewing cluster of tiles as a whole, we have analogue of previous two Orthogonality relation: for P^j ⊂ Pn cluster at pj∈ R^D× Qd, we have

(hcluster_P⁰f, cluster_P1f i = 0 ⇐=S P⁰∩S

P¹= ∅

|hcluster^∗_P0f, cluster^∗_P1f i| . 2⁻ⁿ(1 + distancep₀,p₁)⁻. (8.3.1)

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Combining what have been learned, a reasonable strategy to solve the puzzle would be to organize Pⁿ into the following two ”good” configurations:

• Sparse Parts: P ⊂ Pnhas few overlaps on R^D×Qd, and Orthogonality gives strong enough control. (Details are presented in7)

• Cluster Parts: P ⊂ Pn consists of multiple clusters but clusters are 2^Cn apart on R^D× Qd with C 1. By combining both Orthogonality and Multi-Resolution Analysis, we can apply Cotlar-Stein Lemma and arrive at a suitable control. (Details are presented in8).

2.3 Combinatorial Wizardry and Analytic Magecraft

Now, to systematically extract those good configurations from Pⁿ, we follow both [Lie20] and [Zor19] , which follow Charles Fefferman’s idea in [Fef73]. To elaborate, we equip Pⁿwith an ”order-like” relation to reflect their ”incidental properties”. Consequently, both sparse parts and cluster parts have alternate interpretations:

• Sparse Parts: Collections of Anti-Chains

• Cluster Parts: Collections of Convex Sets

Therefore, through some Combinatorial methods devised by Fefferman, we can extract the desired configurations. (Details in5.3.)

Still, the original argument in [Fef73] has no control over how ”high” clusters stack. The author isolates those who stack too high and proves that they have

”small supports”, which is why ”Exceptional Sets” arise in [Fef73]. This prevents us from finer estimate and direct L²→ L² bound.

One of the innovation in [Lie20] is the clever use of John-Nirenberg inequality. The arguments guarantee that ”higher clusters” has ”smaller supports”. That is, instead of stacking like Jenga, the clusters stack like Eiffel Tower. Consequently, Lie eliminated the use of Exceptional Sets and derived L²→ L² bound directly. (Details in6.3.)

On the other hand, Zorin-Kranich simplified the argument and put additional steps to make the system more compatible with certain ”temporal dilation”. (Details in6.4.)

Finally, to acquire full L^p→ L^pbound, we modify Lie’s argument on sparse parts with the language in [LN15] and adopt Zorin-Kranich’s treatment on cluster parts. To be more specific, we first derive p-bounds insensitive to density:

• Sparse Parts: We resort to pointwise sparse dominance on sparse parts.

• Cluster Parts: We use the Multi-Resolution Analysis on clusters to derive ”localized estimate” and the extrapolation method adopted by Bateman in [BT13] to acquire L^p,1→ L^p,∞bound. (Detail in8.7.) To complete the argument, we interpolate to spread the 2⁻ⁿfactor to L^p^θ → L^p^θ bound and use the geometric decay on density to sum everything up.

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3 Tools and Facts

In this section, we establish some tools and some useful facts without proof.

For starters, we borrow part of the setting and language in [Zor19] and [SW01] to quantify the effect of polynomial phases on behavior of oscillatory integrals.

Next, we follow the setting in [LN15] and sum up some useful facts about sparse systems.

At the end of the section, we introduce our modified settings and explain how it relates to the original settings and why the change of the formulation in [Zor19] may be necessary to generalize the result in [Lie20].

Remark. Throughout this thesis, we will sometimes suppress the dependence on κ, κ^∗, D, d within the ., , ⊂∼ relation.

3.1 Local Oscillation of Polynomial

To apply Cotlar-Stein Lemma, we expect the need for an estimate as the following:

q ∈ Qd, ψ ∈ L⁰(measurable function) =⇒

ˆ

e^iqψdµ

.

D,d Oscillation of ψ,q

on suppψ

?

Indeed, when d = 1, Riemann–Lebesgue Lemma gives us qualitative descrip- tion: the higher the oscillation, the greater the cancellation. This motivates the need to quantify the oscillation of q within the support of ψ. However, to simplify the matters, we model the support as cubes, and we, therefore, need some related terminology:

Definition 3.1.1 (Attributes of a cube I ⊂ R^D).

• cI ∈ R^D denotes the center of mass of I.

• `I denotes the side-length of I.

• |I| := `ID denotes the D-volume of I.

In short, I := cI+ Ì[−1/2, 1/2)^D= cI+ [−Ì/2, Ì/2)^D. Definition 3.1.2 (Temporal Dilation).

∀C ∈ R+, CI := cI+ C`I[−1/2, 1/2)^D= cI+

−C`I

2 ,C`I

2

^D . Now, we define a weaker form of ”⊂”. Given I, J ⊂ R^D be cubes, Definition 3.1.3 (Roughly Contain).

I ⊂∼ J ⇐⇒ ∃C ∈ R⁺ prescribed, s.t. I ⊂ CJ

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Finally, we characterize the local oscillation of q ∈ Qd on cube.

Definition 3.1.4 (Seminorm on Q_d [Zor19](4.1.5.)).

kqkI := sup

x,y∈I

|q(x) − q(y)| .

As an immediate result, since Q_d is a finite dimensional vector space, all non-trivial(vanishing only on constant) seminorms are equivalent. Therefore, we may unambiguously assign a topology generated by seminorm on Q_d. Still, for our purpose, we need quantitative controls:

Properties 3.1.5 (Embedding Inequality [Zor19]Lemma 4.1.6.).

I ⊂∼ J =⇒ `J

`I

kqkI .

D,d

kqkJ .

D,d

`J

`I

^d kqkI,

Such estimate would become important as we do Multi-Resolution Analysis.

3.2 Van der Corput Estimate

Continuing previous settings,

Properties 3.2.1 ([Zor19]Lemma 4.6.1. [SW01]Proposition 2.1.).

∀ψ ∈ L⁰, suppψ ⊂ I =⇒

ˆ

e^iqψdµ .

D,d

sup

k∆k

`I <hkqkIi^1/d

kψ − τ∆ψk_L1

. sup

k∆k

`I <hkqk_Ii^1/d

kψ − τ∆ψkL^∞|I|.

where h·i :=_1+|·|¹ and τ_∆ψ(·) := ψ(· − ∆).

As a immediate corollary, we have a version designed for partition of unity:

For generic ψ ∈ L⁰, δ > 1, I ⊂ R^Dbe cube, we consider a fragment of partition of unity located around I. That is,

χ ∈ C_c^∞s.t.







|χ| .

δ

χδI

k∇χk .

δ

χ_δI/`_I, and we have

Corollary 3.2.1.1.

ˆ

χe^iqψdµ .

D,d,δ

|I|











hkqk_Ii^1/dkψk_L∞((1+2δ)I) Height of ψ +

sup

|∆|

`I <hkqkIi^1/d

kψ − τ∆ψk_L∞(δI) Oscillation of ψ.

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3.3 Sparse Language and Ambient System

The Sparse System we refer to is a sub-system of a 2^κ-adic System satisfying certain properties. For our purpose, we do not work under usual Dyadic Sys- tem. Yet, all the language in [LN15] can be easily converted. For starters, we construct our ambient system:

Definition 3.3.1 (Standard 2^κ-adic System hD, ⊂i).

D :=

G

s∈Z

D^s, where D^s:=2^sκ ζ + [0, 1)^D

be cube | ζ ∈ Z^D .

We equip D with ⊂ as partial order and, for I ⊂ D, define:

(M I := {I ∈ I | @J ∈ I s.t. I ( J} maximal elements I^⊂:= {I ∈ D | ∃J ∈ I s.t. I ⊂ J} downward envelope.

Also, we denote the parent(immediate predecessor) of I ∈ D as bI ∈ D.

Now, given S ⊂ D, 1 ≤ Λ, we call S a Sparse System if it satisfies either of the following equivalent([LN15] 6.1.) conditions:

Definition 3.3.2 (Λ-Carleson Condition [LN15] Definition 6.2.).

S is Λ-Carleson ⇐⇒ ∀J ∈ S(or equivalently, D), X

I∈S, I⊂J

|I| ≤ Λ|J |.

Definition 3.3.3 (Λ⁻¹-Sparse Condition [LN15] Definition 6.1.).

S is Λ⁻¹-Sparse ⇐⇒ ∀I ∈ S, ∃E^I ⊂ I measurable s.t.

(|I| ≤ Λ|EI|

E_Is are disjoint . With basic terminology established, we provide the following two constructions. Given D^ω→ R^(·) +, S Λ-Carleson, we construct







Mω(·) := sup

·∈I∈D

ωI

S_S,ω(·) :=X

I∈S

ωIχI(·) . Through Definition 3.3.3., we relate the two constructions:

Lemma 3.3.4 (Sparse-Maximal Dominance).

|hS_S,ω, f i| ≤X

I∈S

ωI|hχI, f i|

≤X

I∈S

|I|ωI I

|f |dµ ≤X

I∈S

Λ|EI|ωI I

|f |dµ

≤ΛX

I∈S

ˆ

E_I

MωM f dµ ≤ ΛhMω, M f i

=⇒kS_S,ωk_Lp.

p

ΛkM_ωk_Lp.

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3.4 Modified Settings

We introduce a smoothed-out but scale-discretized version of T_∗ and C_d∗, which would become major tools later on. For our purpose, we

1. Prescribe nD:= d2√

D + 1e ∈ N, κ

D,d1, δ

D,d2^−κ, where the values of 2^κ∈ N, δ ∈ R+ would be made clear in the subsequent sections.

2. Fix χ ∈ C_c^∞ satisfying:

χ_(n_D_+δ)[−1,1]D ≤ χ ≤ χ_(n_D₊₂−κ−δ)[−1,1]^D. 3. Define φ(·) := χ(2^−κ·) − χ(·) ∈ C_c^∞. Note that:

suppφ ⊂ (−nD2^κ− 1, nD2^κ+ 1)^D\[−nD, nD]^D.

As a result, certain shifts Sh := {z ∈ Z | nD≤ |z| ≤ n_D2^κ+ 1}^Dyield

x ∈ [0, 1)^D =⇒

(suppφ(x − ·)

suppφ(· − x) ⊂ G

ξ∈Sh

ξ + [0, 1)^D,

and, by our constructions,

x, x⁰∈ [0, 1)^D ∧ y ∈ G

ξ∈Sh

ξ + [0, 1)^D =⇒ kx − x⁰k kx − yk ≤

√ D

nD− 1 ≤ 1/2, which is exactly the condition for τ -H¨older Type Control of K. For convenience, we also define for I ∈ D the following collection and set:

ShI := {`Iξ + I ∈ D | ξ ∈ Sh} and I^∗:=G ShI. 4. Decompose K into wavelet-like pieces:

K =X

s∈Z

K_s

where ∀x, y ∈ R^Ds.t. x 6= y

Ks(x, y) := φ 2^−sκ(x − y) K(x, y)

Since Ksinherits the standard kernel properties of K and the support constraint on φ, translation and dilation yield the following three properties:

Properties 3.4.1 (L⁰\Support Control).

x ∈ I ∈ D^s =⇒

(suppK_s(x, ·)

suppKs(·, x) ⊂ I^∗.

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Properties 3.4.2 (L^∞\Size Control).

|Ks| .

D,d

2^−sDκ.

Properties 3.4.3 (τ -H¨older Regularity).

x, x⁰∈ I ∈ Ds =⇒

(|K_s(x, ·) − K_s(x⁰, ·)|

|Ks(·, x) − Ks(·, x⁰)| .

D,d

kx − x⁰k

`I

^τ

|I|⁻¹χ_I^∗(·).

Corollary 3.4.3.1 (Locally τ -H¨older Continuity).

|x − x⁰| . 2^sκ =⇒

(|Ks(x, ·) − Ks(x⁰, ·)|

|K_s(·, x) − K_s(·, x⁰)| .

D,d

(2^−sκkx − x⁰k)^τ2^−sDκ.

Proof. Given |x − x⁰| . 2^sκ, we can always find . 1 cubes Ij ∈ Ds covering the straight line joining x and x⁰ with x_j ∈ I_j on the line, where x = x₀ and x⁰ = x_n, such that:

|K_s(x, ·) − K_s(x⁰, ·)| ≤

n

X

j=1

|K_s(x_k, ·) − K_s(x_k−1, ·)|

.

D,d n

X

j=1

kxk− xk−1k

`_I_j

^τ

|Ij|⁻¹. 2^−sκkx − x⁰k^τ 2^−sDκ.

The dual notion holds similarly.

With such scale decomposition, we may define:

Definition 3.4.4 (Modified Truncated Maximal CZO).

T∗f (·) := sup

s<s

ˆ s

X

s=s

Ks(·, y)f (y)dy .

By tinkering with (s, s, r, R) ∈ Z²× R²+ so that

(n_D2^sκh r nD2^sκh R

, we have:

ˆ

r≤k·−yk<R

K(·, y)f (y)dy − ˆ s

X

s=s

Ks(·, y)f (y)dy

.

D,d

M f (·).

As a result, Properties 3.4.5.

|T_∗f − T_∗f | .

D,d

M f.

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Therefore, the L^p→ L^p behaviors of T_∗and T_∗are identical. Consequently, it is relevant to consider:

Definition 3.4.6.

Cd∗f (·) := sup

q∈Q_d

T∗(e^iqf )(·), and immediately, we have:

Corollary 3.4.6.1.

|C_d∗f − C_d∗f | .

D,d

M f.

Eventually, L^p→ L^pbehavior of Cd∗is governed by Cd∗, and the main result Theorem1.3.2can be reduced to proving:

Theorem 3.4.7.

kCd∗f kL^p .

D,d,p

kf kL^p, ∀p ∈ (1, ∞)

On the other hand, the main result Theorem1.3.3for Singular Integral type operator cannot be derived directly through such method, since, in general:

T f (·) := lim

→0⁺

ˆ

≤k·−yk

K(·, y)f (y)dy 6=X

s∈Z

ˆ

Ks(·, y)f (y)dy,

even if:

∀⁰x ∈ R^D, lim

→0⁺

ˆ

≤kx−yk≤1

K(x, y)dy exists.

Unless, K is, for example, Anti-Symmetric: in Lie’s works [Lie08], [Lie20], D = 1, K(x, y) = _x−y¹ . If we choose χ ∈ C_c^∞ even, we have:

∀s ∈ Z, ˆ

K_s(·, y)dy = 0.

As a result, by using M.V.T. and D.C.T., we have:

X

s<s≤s

ˆ

Ks(·, y)f (y)dy

= ˆ

X

s<s≤s

Ks(·, y) (f (y) − f (·)) dy

s%∞−→

s&−∞

ˆ

K(·, y) (f (y) − f (·)) dy

= p.v.

ˆ

K(·, y)f (y)dy = Hf (·) = T f (·).

In conclusion, for general standard kernel K, we should adopt Zorin-Kranich’s approach in [Zor19].

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4 Decomposition of the Operator

In the section, we provide the rigorous version of the following decomposition:

C˜d∗f (·) ∼X

P

waveletsP∗ e^iq^(·)f (·)χEP(·).

To be more specific, since we’ve established that

kCd∗f kL^p. kf k^L^p⇐= kCd∗f k . kf k^L^p,

we may shift our focus to C_d∗ for the rest of the arguments. Our goal is to reduce C_d∗f into sum and maximum over finite elements, to linearize the operator, and to do the tile decomposition.

4.1 Reduction and Linearization

For starters, we notice that Observation. Qd is separable.

That is, by explicitly enumerating rational coefficient polynomials:

{q ∈ Q[x1][x₂] · · · [x_D] | degq ≤ d} =: {q_n}_n∈N, Fatou’s Lemma and some limiting arguments yield:

Cd∗f (·) = sup

n∈N

T∗(e^iqⁿf )(·)

= sup

n∈Ns<s

ˆ s

X

s=s

K_s(·, y)e^iqⁿ^(y)f (y)dy

←−

as N →∞ max

n≤N

−N ≤s<s≤N

ˆ ^s X

s=s

Ks(·, y)e^iqⁿ^(y)f (y)dy

=: Cd∗,Nf (·)

Finally, by M.C.T.,

kCd∗f kL^p= sup

N ∈N

kCd∗,Nf kL^p.

Consequently, we only need to acquire bounds on Cd∗,Nf independent of N . Indeed, C_d∗,Nf is a sum and a maximum over finite elements. As a result, we can do an elementary stopping time argument to linearize C_d∗,Nf :

∀N ∈ N, ∃









 R^D

s_(·)

−→ {−N, −N + 1, · · · , N − 1}

R^D

s_(·)

−→ {−N + 1, · · · , N − 1, N } R^D

q_(·)

−→ {qn}^N_n=1

simple and measurable

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such that

Cd∗,Nf (·) =

ˆ ^s^(·) X

s=s_(·)

Ks(·, y)e^iq^(·)^(y)f (y)dy

That is, regardless of the choice of N ∈ N, the problem reduces to analyze the following form of linear operator:

Lf (·) :=

s_(·)

X

s=s_(·)

ˆ

Ks(·, y)e^iq^(·)^(y)f (y)dy,

where s_(·), s_(·), q_(·) are simple measurable functions.

4.2 Tile Decomposition and Trivial Estimate

To proceed with our 3-step decomposition schemes, we first need to refine the following relation:

2^s∼ diameter of I ∼ diameter of ω⁻¹.

For our purpose, we adjust the above statement to our modified settings:

• 2^sκ is the actual scaling that works well with our analysis.

• I ⊂ R^Dis an element chosen from D^s(Standard 2^κ-adic System) to match the 2^sκ-scale.

• ω ⊂ Qdwill be chosen from D^∗I, a Qd-tiling (assumed to exist) that respects the oscillation of polynomials on I and the Uncertainty Principle:

q, q⁰∈ ω =⇒ kq − q⁰kI . 1

Notice that, by the definition of Dsand theEmbedding Inequality, dimensional analysis yields

(2^sκ= `I h

Ddiameter of I

I’s and ω’s ”diameters” are scale-reversed

Naturally, we follow our convention and denote the natural scaling s as sI×ω. For now, we shall postpone the construction of D^∗I and complete the decomposition first:

Lf (·) =

s

X

s=s

X

I∈Ds

X

ω∈D^∗_I

ˆ

K_s(·, y)e^iq^(·)^(y)f (y)dy · χ_E_I×ω(·),

where







s := max

x∈R^D

s_x s := min

x∈R^D

s_x and EI×ω:= {x ∈ I | qx∈ ω ∧ s_x≤ sI×ω≤ sx} . To further simplify the notation, we shall organize the I-ω parings and define:

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Definition 4.2.1 (Tile System).

D :=˜

s

G

s=s

G

I∈Ds

{I × ω ⊂ R^D× Qd | ω ∈ D^∗I}

Definition 4.2.2 (A piece associated to P ∈ ˜D).

LPf (·) :=

ˆ

Ks_P(·, y)e^iq^(·)^(y)f (y)dy · χE_P(·) Immediately, support and size controls yield:

Properties 4.2.3 (Single tile estimate).







|LPf | .

D,d

2^κD|f |I˜PχE_P

|L^∗_Pf | .

D,d

kf k_L1(EP)

|IP| χI˜P

, where ˜I := nD2^κ+1+ 3 I ⊃ I^∗.

Through direct computation, we also have:

(kL_Pf k_L1 . |f |I^˜P|E_P| . A(P )kf kL¹( ˜IP)

kLPf k_L^∞ . |f |I^˜_P ≤ kf k_L∞( ˜I_P), (where A(P ) := ^|E_|I^P^|

P|) and, through interpolation:

Corollary 4.2.3.1 (Trivial Estimate).

kLPf kL^p .

κ,D,d,p

A(P )^1/pkf k_Lp( ˜IP)

On the other hand, given P ⊂ ˜D, we set:

Definition 4.2.4.

L_Pf :=X

P ∈P

LPf

Eventually, we have the succinct expression:

Lf = LD˜f := X

P ∈˜D

L_Pf

with each piece behaving ”nicely”. Moreover, since

• f ∈ Cc^∞ has compact support,

• s(·), s_(·), q_(·) have finite ranges,

the sum only consists of finitely many non-zero terms. As a result, we may freely rearrange and reorganize the sum.

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4.3 Adaptive Christ Grid Construction

Before we construct D^∗I, let us list what we expect from the construction:

• D^∗I tiles Q_d, and, when viewed as in hQ_d, k · k_Ii, every piece in D^∗I contains and is contained in a ball with radiush 1.

• Given J ⊂ I, (ω, ω⁰) ∈ D^∗I× D^∗J, we have either ω ∩ ω⁰= ∅ or ω ⊂ ω⁰. In short, we would like to have a hyper-adic system on Qd. To do so, Zorin- Kranich follows Michael Christ’s idea on constructing dyadic system on space of homogeneous type. However, the construction would be much easier since we only need to consider I ∈ D^s, where s ≤ s ≤ s. Essentially, we can work our ways down from the top scale s. By constructing the finest layer first, the rest of the arguments become finding the correct ways to group the pieces together.

For starters, we prescribe κ^∗

D,d1 and, by using theEmbedding Inequality, find κ

D,d

1 such that, given J ⊂ I be cubes and q ∈ Q_d, we have:

`J ≤ 2^−κ`I =⇒ kqkJ≤ 2^−κ^∗kqkI. We now set ς :=₂_κ∗¹₋₁ and proceed inductively as follows:

(s = s − 0): For all I ∈ Ds,

(a) we select a maximal collection of polynomials Q_I ⊂ Q_d such that

∀q, q⁰∈ QI, q 6= q⁰ =⇒ kq − q⁰kI ≥ 1.

Due to maximality,

( Q_d ⊂S

q∈QIB_I(q, 1)

∀q, q⁰ ∈ QI, q 6= q⁰ =⇒ BI(q, 1/2) ∩ BI(q⁰, 1/2) = ∅, where BI(c, r) := {q ∈ Qd | kq − ckI < r}.

(b) we construct the Qd-tiling D^∗I inductively with each piece assigned a center. That is, ∃D^∗I

c_(·) ω_(·)

QI such that, for all ω ∈ D^∗I,

BI(cω, 1/2 − ς) ⊂ BI(cω, 1/2) ⊂ ω ⊂ BI(cω, 1) ⊂ BI(cω, 1 + ς).

(s > s − k): Suppose the construction be completed so that:

(a) for all I ∈ Ds, we have a Q_d-tiling D^∗I.

(b) we assign for each piece in D^∗I a unique center: ∃D^∗I c_(·) ω_(·)

QI, where

ω ∈ D^∗I =⇒ B_I(c_ω, 1/2 − ς) ⊂ ω ⊂ B_I(c_ω, 1 + ς) .

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(s = s − k): Given I ∈ D^s+1, for all J ∈ D^s∩ 2^I,

(a) we select a maximal collection of polynomials QJ⊂ QI such that

∀q, q⁰ ∈ QJ, q 6= q⁰ =⇒ kq − q⁰kJ ≥ 1.

Due to maximality,

( Q_I ⊂S

q∈QJB_J(q, 1)

∀q, q⁰∈ QJ, q 6= q⁰ =⇒ BJ(q, 1/2) ∩ BJ(q⁰, 1/2) = ∅, (b) we construct inductively a partition on Q_I indexed by Q_J:

{Chq}q∈Q_J where ∀q ∈ QJ, BJ(q, 1/2)∩QI ⊂ Chq ⊂ BJ(q, 1)∩QI. (c) we define ω_(·), by setting:

D^∗J:= {ω_q}q∈QJ, where ω_q := G

q⁰∈Chq

ω_q⁰,

with D^∗J c_(·)

→ QJ defined naturally. Essentially, ∀q ∈ QJ, {ωq⁰}q⁰∈Chq

is the collection of children of ωq.

(d) we characterize the size of each piece in D^∗J: pick q ∈ QJ,

• Exterior:

ωq := G

q⁰∈Chq

ωq⁰ ⊂ [

q⁰∈Chq

BI(q⁰, 1 + ς)

⊂ [

q⁰∈BJ(q,1)

B_J

q⁰,

:ς 2^−κ^∗(1 + ς)

⊂ B_J(q, 1 + ς)

• Interior:

∀q⁰ ∈ BJ(q, 1/2 − ς), ∃!ω⁰ ∈ D^∗I s.t. q⁰ ∈ ω⁰

=⇒ kcω⁰− qkJ≤ kcω⁰− q⁰kJ+ kq⁰− qkJ

< 2^−κ^∗kcω⁰− q⁰kI + 1/2 − ς

<

:ς

2^−κ^∗(1 + ς) + 1/2 − ς = 1/2

=⇒ c_ω0 ∈ Chq =⇒ q⁰ ∈ ω⁰⊂ ωq =⇒ B_J(q, 1/2 − ς) ⊂ ω_q (s ≤ s ≤ s): In conclusion, we have:

• for every I ∈ Ds, D^∗I tiles Q_d (that is,F

D^∗I = Q_d) and ω ∈ D^∗I =⇒ BI(cω, 1/2 − ς) ⊂ ω ⊂ BI(cω, 1 + ς) .

• for all I, J ∈

s

G

s=s

Ds, if J ⊂ I, then, for any (ω, ω⁰) ∈ D^∗I× D^∗J, we, by our grouping construction, have either ω ∩ ω⁰= ∅ or ω ⊂ ω⁰. Notice that, by setting κ^∗

D,d1, we have 0 < ς

D,d1.

This completes the construction.

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5 From Incidental Geometry to Order Theory and Combinatorics

Organizing tiles is essentially an incidental geometric problem. However, due to the hyper-adic properties of ˜D, we can equip ˜

D an order structure to suitably represent its incidental behavior. As a result, we can treat the order theoretical counterpart with some combinatorial tricks.

5.1 Conversion and Basic Operations

We start with some observations: given I, J ∈ D,

• either I ∩ J = ∅

• or I ⊂ J ∨ I ⊃ J and, thus, for any (ω, ω⁰) ∈ D^∗I× D^∗J, – either ω ∩ ω⁰= ∅

– or ω ⊃ ω⁰ ∨ ω ⊂ ω⁰ respectively.

This motivates the following definition:

Definition 5.1.1 (D ˜D, E E

).

∀P, P⁰∈ ˜D, P E P⁰ ⇐⇒ IP ⊂ IP⁰∧ ωP ⊃ ωP⁰. For strict inequality, we write C instead.

We see that E indeed defines a partial order on ˜D. Moreover, it reflects the incidental properties precisely:

∀P, P⁰∈ ˜D, EP ∩ E_P⁰ = ∅ ⇐= P ∩ P⁰ = ∅ ⇐⇒ P, P⁰ are E -incomparable.

As a result, to extract sparse parts(E-anti-chains), we heavily rely on the following operations:

Definition 5.1.2 (Maximal and minimal elements).

∀P ⊂ ˜D,

(M P := {P ∈ P | @P⁰ ∈ P s.t. P C P⁰} mP := {P ∈ P | @P⁰ ∈ P s.t. P⁰C P }.

We also define the iterated versions:

∀k ∈ N,

(Mk+1P := M (P \ MkP) mk+1P := m (P \ mkP) .

Notice that, by construction, both MkP and mkP are E-anti-chains.

On the other hand, for cluster parts, we shall define:

多項式調變不變奇異積分算子

Department of Mathematics College of Science

National Taiwan University Master Thesis

摘要

關鍵詞：時頻分析、多重解析度分析、CZ算子、稀疏壓制、TT*-T*T方法

誌謝

作者感謝指導教授開明的態度以及十足的耐心，促成這漫長的計畫以及深

度的細部探討。其次作者感謝與多位同事日常討論所激發的靈感，造就了諸多

論證的精進。最後，作者想感謝家人以及身邊每位關心作者的人。

Polynomial Modulation Invariant Singular Integral Operators

Yung-Chang Hsu Feb. 2021

Contents

1 Introduction

1.1 Basic Notions

1.2 Motivation

1.3 Main Result

2 Mathematical Jigsaw Puzzle

2.1 Cut out the Pieces

2.2 Find Good Configurations

2.3 Combinatorial Wizardry and Analytic Magecraft

3 Tools and Facts

3.1 Local Oscillation of Polynomial

3.2 Van der Corput Estimate

3.3 Sparse Language and Ambient System

3.4 Modified Settings

4 Decomposition of the Operator

4.1 Reduction and Linearization

4.2 Tile Decomposition and Trivial Estimate

4.3 Adaptive Christ Grid Construction

5 From Incidental Geometry to Order Theory and Combinatorics

5.1 Conversion and Basic Operations

關鍵詞：時頻分析、多重解析度分析、CZ算子、稀疏壓制、TT-TT方法