ON THE WAVELET SCATTERING TRANSFORM AND ITS GENERALIZATIONS

By

Albert Chua

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics—Doctor of Philosophy

2023

ABSTRACT

In this thesis, we look into generalizations of Mallat’s wavelet scattering transform. In the second

chapter, we generalize finite depth wavelet scattering transforms, which we formulate as L𝑞 (R𝑛)

norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive

nonlinearities. We then provide norms for these operators, prove that these operators are well-

defined, and are Lipschitz continuous to the action of 𝐶2 diffeomorphisms in specific cases;

additionally, we extend our results to formulate an operator invariant to the action of rotations

𝑅 ∈ SO(𝑛) and an operator that is equivariant to the action of rotations of 𝑅 ∈ SO(𝑛). In the

third and fourth chapters, we generalize our results to stochastic process and signals on compact

manifolds, respectively.

ACKNOWLEDGEMENTS

First, and most importantly, I would like to thank my advisor, Yang Yang, for all the support that

he has provided me during the one and a half years we have worked with each other. In spite of

the fact that I always try "interesting" ideas for my research, he has never been dismissive, and has

been very supportive by providing help whenever he can. I’ll always appreciate the support he has

provided me.

I also would like to thank my committee members, Jianliang Qian, Ekaterina Rapinchuk, and

Rongrong Wang for all the support they have provided me thoughout my time at MSU. Without

their thoughtful suggestions for this thesis, it would be much worse condition compared to now.

I also appreciate the help Michael Perlmutter, who was one of the first people to provide edits to

what became the journal version of Chapter 2 of this thesis. I definitely would not have caught most

of those errors myself. I also appreciate his advice for how/where to publish the paper, because I

wasn’t really sure of what to do with it at the time.

I would like to also Anna Little, who was one of the coauthors on the journal version of Chapter

2 in this thesis. She provided a lot of help with foundational parts of the work, which made my life

a lot easier. I also appreciate the time she took to help me through the review process for the paper,

which I was completely unprepared for on my own.

Matt Hirn, who was my advisor during the third and fourth years of my PhD, is also someone I

would like to thank. He was the one who introduced me to the scattering transform, and helped me

while I started my research on nonwindowed scattering transforms. Despite all the trouble I had at

first, he was very patient with me and helped me with any trouble I had. I appreciate all the proofs

he checked, especially all the "proofs" with mistakes that he caught.

Lastly, I would like to thank anyone who I have missed on this list. I don’t think I could have

finished this thesis without the help of many people in my life.

iii

TABLE OF CONTENTS

CHAPTER 1

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.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
1.1 Notation .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
1.2 Machine Learning and Model Fitting . . . . . . . . . . . . . . . . . . . . . .
1.3 Background On Convolutional Neural Networks . . . . . . . . . . . . . . . . .
Invariance, Equivariance, Stability, Frequency Representations, and Machine
1.4
Learning .
1.5 Wavelets .
.
1.6 Scattering Transforms .
1.7 Contributions .

3
5
7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
1
1
2

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CHAPTER 2

.

.

.

GENERALIZING THE NONWINDOWED SCATTERING TRANS-
FORM .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 16
. . . . . . . . . . . . . . . . . . . . . . 16
2.1 Fourier Transforms and Hardy Spaces
. . . . . . . . . . . . . . . . . . . . 17
2.2 Wavelet Scattering is a Bounded Operator
2.3 Stability to Dilations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Stability to Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
. 54
2.5 Equivariance and Invariance to Rotations . . . . . . . . . . . . . . . . . . . .

CHAPTER 3

. .

. 60
EXPECTED SCATTERING TRANSFORMS . . . . . . . . . . . . . .
. 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Background . .
3.2 Wavelet Transforms for Stochastic Processes . . . . . . . . . . . . . . . . . .
. 60
3.3 Scattering Moments and the Expected Scattering Transform . . . . . . . . . . . 61
3.4 The Expected Scattering Transform When 𝑞 = 2 . . . . . . . . . . . . . . . . . 62
3.5 The Expected Scattering Transform When 1 < 𝑞 < 2 . . . . . . . . . . . . . . 66

CHAPTER 4

NONWINDOWED SCATTERING ON COMPACT RIEMANNIAN
MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
. 69
4.1 Notation for Scattering on Manifolds . . . . . . . . . . . . . . . . . . . . . .
4.2 Spectral Filters and the Geometric Wavelet Transform . . . . . . . . . . . . .
. 70
4.3 The Geometric Scattering Transform . . . . . . . . . . . . . . . . . . . . . . . 72
. 74
4.4 Generalizing Geometric Scattering Transforms

. . . . . . . . . . . . . . . .

CHAPTER 5

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

BIBLIOGRAPHY .

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. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 79

iv

CHAPTER 1

INTRODUCTION

1.1 Notation

Set R+ to be the positive real numbers, i.e. R+ := (0, ∞). The gradient of a function 𝑓 : R𝑛 → C
is given by ∇ 𝑓 , the Jacobian of a function 𝑓 : R𝑛 → R𝑚 is given by 𝐷 𝑓 , and the Hessian is given by
(cid:105) 1/𝑞

𝐷2 𝑓 . For 1 ≤ 𝑞 < ∞, the L𝑞 (R𝑛) norm of a function 𝑓 : R𝑛 → C is ∥ 𝑓 ∥𝑞 :=

(cid:104)∫

.

R𝑛 | 𝑓 (𝑥)|𝑞 𝑑𝑥

When 𝑞 = ∞, ∥ 𝑓 ∥∞ := ess sup| 𝑓 |. We will also use the notation, ∥Δ 𝑓 ∥∞ = sup𝑥,𝑦∈R𝑑 | 𝑓 (𝑥) − 𝑓 (𝑦)|,

for the first two chapters of this thesis (which should not be mistaken for applying a Laplacian

operator). Greek letters with a vector symbol, such as (cid:174)𝛼 = (𝛼1, · · · , 𝛼𝑛), will be a multi-index of

nonnegative integers; additionally, we write | (cid:174)𝛼| = 𝛼1 + · · · + 𝛼𝑛, and the usage will be clear from
context. The operator 𝐷 (cid:174)𝛼 is a multi-index of derivatives: 𝐷 (cid:174)𝛼 𝑓 =

𝑓 . For integer 𝑠 ≥ 0,

𝜕 | (cid:174)𝛼|
𝜕𝑥 𝛼
···𝜕𝑥 𝛼𝑛
1
𝑛
1

we define the function space H𝑠 (R𝑛) = { 𝑓 ∈ L2(R𝑛) : 𝐷 (cid:174)𝛼 𝑓 ∈ L2(R𝑛) for | (cid:174)𝛼| ≤ 𝑠}.

1.2 Machine Learning and Model Fitting

The following material is based on [1]. A functional perspective on supervised learning is the

following. Suppose we have a set of data that is split into a training set 𝑇 = {(𝑥𝑖, 𝑦𝑖)}𝑁

𝑖=1, which has
𝑖=1. Our goal is to find a model 𝐹𝜃, parameterized by a set
of weights 𝜃 ∈ R𝑛, that best fits the data with respect to some metric (i.e. mean squared loss). To

known data {𝑥𝑖} ⊂ X and labels {𝑦𝑖}𝑁

check if our model 𝐹𝜃 actually fits the data, we are given a set of test points 𝑇test, which are only

accessible for evaluating the fit of 𝐹𝜃.

Define the set of all possible models F as

F = { 𝑓𝜃 (𝑥) : 𝜃 ∈ R𝑛},

where each 𝑓𝜃 is a model parameterized by weights 𝜃 ∈ R𝑛. One needs to narrow down the

search space by choosing an appropriate model to fit the data. One such instance is when one has

prior knowledge of the distribution of data. For example, consider linear regression; suppose that

{𝑥𝑖}𝑁
𝑖=1

⊂ R𝑛 and 𝑦𝑖 ⊂ R with 𝑦𝑖 = 𝑤𝑇 𝑥𝑖 + 𝜀, where 𝑤 ∈ R𝑛 is a set of unknown weights and 𝜀 is a

small noise. Lastly, assume that we want to find a representation that minimizes the mean squared

1

error:

𝑁
∑︁

𝑖=1

( 𝑓𝜃 (𝑥𝑖) − 𝑦𝑖)2.

At this point, it is natural to restrict the set of functions to have the following representation:

𝑓𝜃 (𝑥) = 𝜃𝑇 𝑥,

𝜃 ∈ R𝑛.

Note that this example is relatively simple. For more complex representations, such as images, one

needs to consider more sophisticated representations. Over the past two decades, convolutional

neural networks have show remarkable success for image recognition tasks. For example, [2, 3,

4, 5] have gradually redefined state-of-the art on benchmark datasets in the 2010s. However, the

mechanisms behind how they work have not been fully understood until recently [6, 7, 8, 9, 10].

1.3 Background On Convolutional Neural Networks

Before we provide more discussion about invariants in machine learning, we will discuss the

architecture for convolutional neural networks.

Consider two discrete functions: 𝑎1 : Z → R and 𝑏1 : Z → R. Practitioners in deep learning

generally define the convolution (which is cross-correlation) as

(𝑎1 ∗ 𝑏1) (𝑖) =

∑︁

𝑗 ∈Z

𝑎1(𝑖 + 𝑗)𝑏1( 𝑗).

(1.1)

More generally, we can assume that we have two dimensional functions 𝑎2 : Z2 → R and

𝑏2 : Z2 → R. The two dimensional convolution is given by

(𝑎2 ∗ 𝑏2)(𝑖1, 𝑖2) =

∑︁

( 𝑗1, 𝑗2)∈Z2

𝑎2(𝑖1 + 𝑗1, 𝑖2 + 𝑗2)𝑏2( 𝑗1, 𝑗2).

(1.2)

This is the first building block for convolution operations similar to the operations seen in deep

learning libraries, such as "Conv2d" in PyTorch. However, in practice, these operations generally

are implemented with finite filters rather than infinite filters like above.

To construct a full Conv2d layer, suppose that we have a set of 𝑁1 functions, and the goal is to

get a representation with 𝑁2 functions via a set of convolutions. Define a set of functions {𝐹𝑛1,𝑛2 }

2

with indexes 1 ≤ 𝑛1 ≤ 𝑁1 and 1 ≤ 𝑛2 ≤ 𝑁2. The Conv2d layer can be mathematically expressed as

𝐶 ( 𝑓 ) =

𝑁1∑︁

𝑛1=1

𝐹𝑛1,𝑛2 ∗ 𝑓 .

(1.3)

After applying 𝐶, a nonlinearity is applied to each entry of the result, and some form of subsampling

is done to reduce the data necessary for the representation. A convolutional neural network, less

formally speaking, is a cascade of applying a Conv2d layer, a nonlinearity, and a subsampling

operator, in that exact order.

1.4

Invariance, Equivariance, Stability, Frequency Representations, and Machine Learning

Let B1,B2 be Banach Spaces and Φ : B1 → B2 be an operator, let 𝑇 : B1 → B1 be an operator.

We say that Φ is invariant to 𝑇 if

Φ𝑇 𝑓 = Φ 𝑓 ,

∀ 𝑓 ∈ B1,

and Φ is a 𝑇-invariant operator. Similarly, for 𝑇 : B1 → B2, for we say that Φ is equivariant with

respect to 𝑇 if

Φ𝑇 𝑓 = 𝑇Φ 𝑓 ,

∀ 𝑓 ∈ B1.

Similar to the regression example, CNNs restrict the the possible set of models we consider.

With respect to images, convolution has two properties that are helpful for image recognition tasks:

• Convolution is inherently a local operation and depends on neighboring pixels. That is to

say, we utilize the underlying geometry of an image.

• Convolution is equivariant with respect to translation. In other words, translating a function

and translating a function after convolution yield the same output.

However, it is not necessarily useful to have translation equivariance. Suppose we have the

following two tasks:

• Determine if a cat is in the picture.

• Determine where the cat is in the picture.

For the first task, the location of the cat does not matter, so translating the cat in the picture is

irrelevant. Thus, we would like a representation that is invariant to translation. On the other hand,

3

in the second task, to keep track of the location of the cat, we would like a representation that is

equivariant with respect to translations. This example illustrates the following point. Using relevant

information about our task is a way of restricting down the search space for possible models.

Along with some type of invariance or equivariance, stability is also an important property for

our representation. Let 𝐿𝛾 𝑓 (𝑥) = 𝑓 (𝛾−1(𝑥)), where 𝛾(𝑥) := 𝑥 − 𝜏(𝑥) for 𝜏 ∈ 𝐶2(R𝑛) suitably

small. We would like a representation such that

∥Φ 𝑓 − Φ𝐿𝜏 𝑓 ∥B2 ≤ 𝐾 (𝜏) ∥ 𝑓 ∥B1

,

and 𝐾 (𝜏) get smaller as 𝜏 get smaller. The intuition is that small deformations of the signal will

not change the representation too much.

An important aspect of convolutional models is their ability to discern frequency information.

Empirically, high frequency information is important for image recognition. In Figure 1.1, one can

Figure 1.1 Left: Polar Bear. Middle: Low Pass filtering. Right: High Pass filtering.

see that the high frequency information is what allows us to determine that the image is in fact an

image of a polar bear, so it is important that a representation can extract high frequency information

properties. Notably, convolutions are useful for this task because of the convolution theorem.

Lastly, one also needs sufficient model complexity to retain enough meaningful information,

which is a key ingredient of deep convolutional neural networks. For example, notice that using the

representation ∥ 𝑓 ∥2

2 yields a translation invariant operator, but any meaningful information about

the function 𝑓 is lost, including high frequency information.

Since convolutional neural networks learn the best model via optimizing a set of weights, it is

hard to study their mathematical properties. Instead, one can consider a proxy by using unlearned

filters to simplify the analysis. Ideally, the representation should have the following properties:

4

• Has some invariance/equivariance properties.

• Stable to small deformations.

• Keeps meaningful information and is sufficiently complex.

Regarding the third point, choosing an operator with sufficient complexity, invariance, and

stability is not an easy task. For now, consider a simple dilation operator 𝐿𝑐 𝑓 (𝑥) = 𝑓 ((1 − 𝑐)𝑥)

for |𝑐| < 1

2𝑛 . A feasible way to extract more information is via a low pass filtering (e.g. define an
operator 𝐾𝜙 𝑓 = 𝑓 ∗ 𝜙, where ˆ𝜙(𝜔) = 1𝐵𝑅 (0) for some 𝑅 > 0). One can check that for functions 𝑓

such that ˆ𝑓 is supported in 𝐵𝑅 (0), we have

∥ 𝑓 − 𝐿𝑐 𝑓 ∥2

2 = ∥ 𝑓 ∗ 𝜙 − 𝐿𝑐 𝑓 ∗ 𝜙∥2
2

≤ 𝑐2 · 𝐶𝑅 ∥ 𝑓 ∥2
2

for some constant 𝐶𝑅. However, high frequency information is lost because ˆ𝑓 is only supported in

some bounded ball.

To keep high frequency information, a feasible translation invariant operator to consider is the

fourier modulus. However, this operator is not even stable with respect dilations with respect to

the 2-norm. The following informal argument from [11]. Suppose that 𝑓 (𝑥) = 𝑒𝑖𝜉𝑥𝜃 (𝑥), where 𝜃

is regular with fast decay. Then one can prove that

∥| (cid:100)𝐿𝜏 𝑓 | − | ˆ𝑓 |∥2 ≈ |𝑐||𝜉 |∥𝜃 ∥2.

Since 𝜉 is arbitrary, we see that we can choose it so that the Fourier modulus is not stable to

dilations. The main point of these examples is to show that Fourier invariants, which are a natural

choice for a feature extractor, are simply not enough. Even for the most simple class of dilations, we

do not have any stability result that can contain high frequency information. To create an operator

with the properties mentioned above, we consider using wavelets.

1.5 Wavelets

We let 𝜓 ∈ L1(R𝑛) ∩ L2(R𝑛) be a wavelet, which means it is a function that is localized in both

space and frequency and has zero average, i.e.,

∫

R𝑛

𝜓(𝑥) 𝑑𝑢 = 0 .

5

Assume 𝑓 ∈ L2(R𝑛). The continuous wavelet transform W 𝑓 ∈ L2(R𝑛 × R+) is defined as:

∀ (𝑥, 𝜆) ∈ R𝑛 × R+ , W 𝑓 (𝑥, 𝜆) := 𝑓 ∗ 𝜓𝜆 (𝑥) .

Furthermore, if 𝜓 satisfies the following admissibility condition

∫ ∞

0

| (cid:98)𝜓(𝜆𝜔)|2
𝜆

𝑑𝜆 = C𝜓 , ∀ 𝜔 ∈ R𝑛 \ {0} ,

(1.4)

for some C𝜓 > 0, then we will say that 𝜓 is a Littlewood-Paley wavelet for the continuous wavelet

transform. If 𝜓 satisfies (1.4), one can show that the norm W 𝑓 computed with a weighted measure

(𝑑𝑥, 𝑑𝜆/𝜆𝑛+1) on R𝑛 × R+ is well defined:

∥W 𝑓 ∥2

L2 (R𝑛×R+) :=

|W 𝑓 (𝑥, 𝜆)|2 𝑑𝑥

∫ ∞

∫

R𝑛

∫

R𝑛

0
∫ ∞

0
∫ ∞

0

| 𝑓 ∗ 𝜓𝜆 (𝑥)|2 𝑑𝑥

∥ 𝑓 ∗ 𝜓𝜆 ∥2
2

𝑑𝜆
𝜆𝑛+1

.

𝑑𝜆
𝜆𝑛+1
𝑑𝜆
𝜆𝑛+1

=

=

We note, in fact, that one can show:

where

𝛽 =

∥W 𝑓 ∥2

L2 (R𝑛×R+) = 𝛽 · C𝜓 ∥ 𝑓 ∥2

2

1/2 if 𝜓 is real valued

1

if 𝜓 is complex valued





.

.

For a function 𝑓 ∈ L2(R𝑛) we define the dyadic wavelet transform 𝑊 𝑓 ∈ ℓ2(L2(R𝑛)) as

If 𝜓 satisfies

𝑊 𝑓 = (cid:0) 𝑓 ∗ 𝜓 𝑗 (cid:1)

𝑗 ∈Z .

|(cid:98)𝜓(2 𝑗 𝜔)|2 = ˆ𝐶𝜓, ∀𝜔 ∈ R𝑛 \ {0} ,

∑︁

𝑗 ∈Z

(1.5)

(1.6)

for some ˆ𝐶𝜓 > 0, then we will say that 𝜓 is a Littlewood-Paley wavelet for the dyadic wavelet

transform. If 𝜓 satisfies (1.6), one can show that the norm 𝑊 𝑓 given below is well defined:

∥𝑊 𝑓 ∥2

ℓ2 (L2 (R𝑛)) :=

∑︁

𝑗 ∈Z

∥ 𝑓 ∗ 𝜓 𝑗 ∥2
2

.

6

In fact, we have the following norm equivalence:

∥𝑊 𝑓 ∥2

ℓ2 (L2 (R𝑛)) = 𝛽 · ˆ𝐶𝜓 ∥ 𝑓 ∥2

2

,

where 𝛽 is defined in (1.5). Wavelets are an ideal choice because the wavelet transform provides a

decomposition of a function into frequency bins.

1.6 Scattering Transforms

We now introduce the windowed scattering transform, which is a simple model for convolutional

neural network with desirable mathematical properties. Let 𝜙 : R𝑛 → R be a low pass filter

( ˆ𝜙(0) ≠ 0) , 𝜓 : R𝑛 → C a suitable mother wavelet ( ˆ𝜓(0) = 0), and let 𝐺 be a rotation group

and 𝐺+ = 𝐺/{−1, 1}, where 1 is the identity element for the group. Define a set of rotations and

dilations by

and

Λ𝐽 := {𝜆 = 2 𝑗𝑟 : 𝑟 ∈ 𝐺+, 𝑗 > −𝐽} if 𝐽 ≠ ∞

Λ∞ := {2 𝑗𝑟 : 𝑟 ∈ 𝐺+, 𝑗 ∈ Z}.

(1.7)

(1.8)

Let 𝜆 = 2 𝑗𝑟 ∈ Λ𝐽. We further assume that our wavelet satisfies the following unitary frame

condition:

is 𝜓 is a complex wavelet, and

|𝜙(2𝐽𝜔)|2 +

∑︁

𝜆∈Λ𝐽

|𝜓(𝜆−1𝜔)|2 = 1

|𝜙(2𝐽𝜔)|2 +

1
2

∑︁

𝜆∈Λ𝐽

(cid:2)|𝜓(𝜆−1𝜔)|2 + |𝜓(−𝜆−1𝜔)|2(cid:3) = 1

if 𝜓 is a real wavelet.

Consider the operator

𝑈 [𝜆] =

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∫

R𝑛

𝑓 (𝑢)2𝑛 𝑗 𝜓(2 𝑗𝑟 −1(𝑥 − 𝑢)) 𝑑𝑢

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(1.9)

For a tuple of rotations and dilations in Λ𝐽, define a path of length 𝑚 as the tuple 𝑝 := (𝜆1, . . . , 𝜆𝑚)
and let P𝐽 be the set of all finite paths. The scattering propagator for 𝑓 ∈ L2(R𝑛) and 𝑝 ∈ P𝐽 is

𝑈 [ 𝑝] 𝑓 := 𝑈 [𝜆𝑚] · · · 𝑈 [𝜆1] 𝑓 ,

(1.10)

7

which gathers high frequency information via a cascade of wavelet transforms and nonlinearities.

The scattering operator is

𝑆 𝑓 ( 𝑝) =

1
𝜇 𝑝

∫

R𝑛

𝑈 [ 𝑝] 𝑓 (𝑥) 𝑑𝑥

(1.11)

with 𝜇 𝑝 := ∫
[11] define the scattering operator for 𝑓 ∈ L2(R𝑛) and 𝑝 ∈ P𝐽 as

R𝑛 𝑈 [ 𝑝]𝛿(𝑥) 𝑑𝑥. Additionally, to aggregate features similar to pooling, the author of

𝑆𝐽 [ 𝑝] 𝑓 (𝑥) =

∫

R𝑛

𝑈 [ 𝑝] 𝑓 (𝑢)2−𝑛𝐽 𝜙(2−𝐽 (𝑥 − 𝑢)) 𝑑𝑢.

(1.12)

Additionally, the windowed scattering transform is the set of functions

𝑆𝐽 [P𝐽] 𝑓 = {𝑆𝐽 [ 𝑝] 𝑓 } 𝑝∈P𝐽 .

(1.13)

This operator is similar to a convolution neural network because along each path (analogous to

each layer of a convolutional neural network) a convolution, a nonlinearity is applied, and feature

aggregation occurs via the low pass filter. The scattering norm for any set of paths Ω is

∥𝑆𝐽 [Ω] 𝑓 ∥2 =

∑︁

𝑝∈Ω

∥𝑆𝐽 [ 𝑝] 𝑓 ∥2
2

.

(1.14)

Notably, we see that the windowed scattering transform has a structure similar to a convolutional

neural network. Since it is important for a feature extractor to extract high frequency information,

we will provide an informal explanation for how the modulus nonlinearity does this.

Suppose 𝑓 ∈ 𝐿2(R𝑛). Then

(cid:156)( 𝑓 ∗ 𝜓 𝑗 ) (0) = ˆ𝑓 (0) ˆ𝜓 𝑗 (0) = 0,

and assume that 𝜓 is 𝐶∞ without any loss of generality. Assume 𝑓 ∗ 𝜓 𝑗 ≠ 0 on a set of positive

measure. Then

(cid:156)| 𝑓 ∗ 𝜓 𝑗 |(0) =

∫

R𝑛

| 𝑓 ∗ 𝜓 𝑗 |(𝑥) 𝑑𝑥 > 0.

Since | 𝑓 ∗ 𝜓 𝑗 | is continuous, we can find a neighborhood around the origin where |((cid:155)𝑓 ∗ 𝜓 𝑗 ) (𝑥)| is

nonzero. In other words, high frequency information is pushed down to lower frequency bins.

Before we discuss the theoretical properties of scattering transforms, we provide empirical

justification of scattering architectures for feature extraction. First, the seminal paper [12] provided

8

justification for using the windowed scattering transform for small benchmark datasets. From then

on, scattering features have shown competitive results for audio tasks [13, 14, 15] and image tasks

[16, 17]. Adding learning, like in [18, 19], have been shown to help improve performance in

classification tasks as well.

Moving on to theoretical properties of the windowed scattering transform, the windowed scat-

tering transform has the following properties, which are desirable for a feature extractor. The first

property is energy preservation, under strict assumptions on the wavelet.

Theorem 1 ([11]). A scattering wavelet 𝜓 is said to be admissible if there exists 𝜂 ∈ R𝑛 and 𝜌 ≥ 0,

with | ˆ𝜌(𝜔)| ≤ | ˆ𝜙(2𝜔)| and ˆ𝜌(0) = 1, such that the function

satisfies

ˆΨ(𝜔) = |𝜌(𝜔 − 𝜂)|2 −

(cid:16)

1 − | ˆ𝜌(2−𝑘 (𝜔 − 𝜂))|2(cid:17)

𝑘

∞
∑︁

𝑘=1

𝛼 = inf

1≤|𝜔|≤2

∞
∑︁

∑︁

𝑗=−∞

𝑟∈𝐺

ˆΨ(2− 𝑗𝑟 −1𝜔)| ˆ𝜓(2− 𝑗𝑟 −1𝜔)|2 > 0.

(1.15)

(1.16)

If a wavelet is admissible, then ∥𝑆𝐽 [𝑃𝐽] ∥ = ∥ 𝑓 ∥.

The problem with the admissibility condition in above is that there are very few classes of

wavelets that are admissible. The author of [11] mentions an analytic cubic spline Battle-Lemarié

wavelet is admissible in one dimension, but provides no other examples. On a related note, [20]

has shown that scattering coefficients have exponential decay for 𝑛 = 1 under relatively mild

assumptions, but her proof only applies for 𝑛 = 1, which makes the admissibility condition still

necessary for 𝑛 ≥ 2. Additionally, to our knowledge, there are no examples in the literature of

wavelets that satisfy the admissibility condition when 𝑛 > 1.

The second property is that the windowed scattering transform is nonexpansive.

Theorem 2 ([11]). Suppose 𝜓 is an admissible wavelet. For all 𝑓 , ℎ ∈ 𝐿2(R𝑛),

∥𝑆𝐽 [𝑃𝐽] 𝑓 − 𝑆𝐽 [𝑃𝐽]ℎ∥ ≤ ∥ 𝑓 − ℎ∥2.

The third property is an "almost translation invariance" property.

9

Theorem 3 ([11]). Define 𝐿𝑐 𝑓 (𝑢) = 𝑓 (𝑢 − 𝑐). For admissible wavelets,

∥𝑆𝐽 [𝑃𝐽] 𝑓 − 𝑆𝐽 [𝑃𝐽] 𝐿𝑐 𝑓 ∥ = 0.

lim
𝐽→∞

for all 𝑐 ∈ R𝑛 and for all 𝑓 ∈ 𝐿2(R𝑛).

The last property is a deformation stability bound.

Theorem 4 ([11], informal). Let 𝜏 ∈ 𝐶2(R𝑛) and 𝐿𝜏 𝑓 = 𝑓 (𝑢 − 𝜏(𝑢)). For 𝑓 ∈ 𝐿2(R𝑛) and

∥𝐷𝜏∥∞ < 1
2𝑛 ,

∥𝑆𝐽 [𝑃𝐽] 𝐿𝜏 𝑓 − 𝑆𝐽 [𝑃𝐽] 𝑓 ∥ ≤ 𝐾 (𝜏) ∥ 𝑓 ∥2

with 𝐾 (𝜏) → 0 as ∥𝜏∥∞ + ∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞ → 0.

Deformation stability bounds have become a major point of importance in mathematical deep

learning. Since Mallat’s work, other works have tried to find feature extractors with similar

mathematical properties. For example, [21, 22] consider a generalization of the scattering transform

where one uses a general frame instead of a wavelet frame. Another set of related works are [23, 24],

which uses a generalization of gabor frames, called uniform covering frames, as a convolution layer.

Convolutional kernel networks, as seen in [25, 6], also have desirable mathematical properties.

Additionally, rather than working on Euclidean space, a better intrinsic representation can be found

by working on a graph or manifold (e.g. point cloud data); works such as [26, 27, 28, 29, 30] focus

on feature extractors on noneuclidean data. We will provide a preliminary generalization of [28] in

Chapter 4 of this thesis.

Notably, other than [11, 23, 24] all these feature extractors for Euclidean data only provide

stability bounds for bandlimited functions, or the set of functions that satisfy

{ ˆ𝑓

:

ˆ𝑓 has compact support}.

This assumption is reasonable for actual signals because real-world implementation of signals are

implemented on a domain with compact time and frequency support.

The work in [23, 24] makes a slight generalization to (𝜖 − 𝑅) bandlimited functions. Let

𝑄 𝑅 (𝑥) = {𝑦 ∈ R𝑛 : ∥𝑦 − 𝑥∥∞ < 𝑅}.

10

A function 𝑓 ∈ L2(R𝑛) is (𝜖, 𝑅) bandlimited for some 𝜀 ∈ [0, 1) and 𝑅 > 0 if

∥ ˆ𝑓 ∥L2 (𝑄 𝑅 (0)) ≥ (1 − 𝜀) ∥ 𝑓 ∥2.

However, their stability result is slightly weaker because there are terms that are independent of the

deformation in their bound.

To our knowledge, a result similar to Mallat’s stability bound, which does not rely on the

function being bandlimited, does not exist for other feeature extractors in the current literature. An

interesting line of work appears in [31], where one relaxes the assumption on 𝜏 in Theorem 4 from

𝜏 ∈ 𝐶2(R𝑛) to 𝜏 ∈ C1+𝛼 (R𝑛) for 𝛼 ∈ (0, 1). Similar results also apply to our stability bound in

Chapter 2 as well.

1.7 Contributions

Windowed Scattering Transforms are useful when the representation does not need to be rigid.

For example, object detection does not require translation invariance, so a Windowed Scattering

Transform would be appropriate since a smaller choice of 𝐽 would not have coeffiicents that would

be nearly translation invariant. For a task like classification that needs rigid translation invariance,

windowed scattering coefficients are not necessarily the best option. Since the set of functions {𝜙𝐽 }

forms an approximate identity,

lim
𝐽→∞

𝑆[ 𝑝] 𝑓 = lim
𝐽→∞

2𝑛𝐽 ∫

R𝑛

𝑈 [ 𝑝] ( 𝑓 ∗ 𝜙𝐽) (𝑥) 𝑑𝑥 = 𝜙(0) ∥𝑈 [ 𝑝] 𝑓 ∥1.

Here, the norm acts as the global pooling layer instead of a local pooling layer with the low pass

filter. Mallat considered the set of all nonwindowed scattering coefficients, given by 𝑆[𝑃∞] 𝑓 ,

which provides a rigid representation. However, he was not able to provide stability results for the

norm he considered.

We consider a slightly different problem than Mallat did for the nonwindowed scattering

transform. As mentioned before, the nonwindowed scattering transform introduced in [11] was

a collection of L1(R𝑛) norms of various cascades of dyadic wavelet convolutions and modulus

nonlinearities applied to a signal. Here, we extend the definition of the scattering transform to the

11

continuous wavelet transform and for L𝑞 (R𝑛) norms with 𝑞 ∈ [1, 2]. For a continuous dilation

parameter 𝜆 ∈ R+ we define the dilations of 𝜓 as:

∀ 𝜆 ∈ R+ , 𝜓𝜆 (𝑥) := 𝜆−𝑛/2𝜓(𝜆−1𝑥) ,

which preserves the L2(R𝑛) norm of 𝜓:

∥𝜓𝜆 ∥2 = ∥𝜓∥2 , ∀ 𝜆 ∈ R+ .

For the continuous wavelet transform, the one layer wavelet scattering transform with L𝑞 (R𝑛) norm

is the function 𝑆cont,𝑞 : R+ → R defined as:

∀ 𝜆 ∈ R+ ,

𝑆cont,𝑞 𝑓 (𝜆) := ∥ 𝑓 ∗ 𝜓𝜆 ∥𝑞 .

(1.17)

For a dyadic dilation parameter 𝑗 ∈ Z we define dilations of 𝜓 as:

∀ 𝑗 ∈ Z , 𝜓 𝑗 (𝑥) = 2−𝑛 𝑗 𝜓(2− 𝑗 𝑥) ,

which preserves the L1(R𝑛) norm of 𝜓:

∥𝜓 𝑗 ∥1 = ∥𝜓∥1 , ∀ 𝑗 ∈ Z .

The one layer wavelet scattering transform for the dyadic wavelet transform is the function 𝑆dyad,𝑞 𝑓 :

Z → R defined as:

∀ 𝑗 ∈ Z ,

𝑆dyad,𝑞 𝑓 ( 𝑗) := ∥ 𝑓 ∗ 𝜓 𝑗 ∥𝑞 .

(1.18)

More generally, the 𝑚-layer wavelet scattering transforms 𝑆𝑚

cont,𝑞 𝑓

Z𝑚 → R are defined as

: R𝑚

+ → R and 𝑆𝑚

dyad,𝑞 𝑓

:

𝑆𝑚
cont,𝑞 𝑓 (𝜆1, . . . , 𝜆𝑚) := ∥|| 𝑓 ∗ 𝜓𝜆1 | ∗ 𝜓𝜆2 | ∗ · · · | ∗ 𝜓𝜆𝑚 ∥𝑞 ,

𝑆𝑚
dyad,𝑞 𝑓 ( 𝑗1, . . . , 𝑗𝑚) := ∥|| 𝑓 ∗ 𝜓 𝑗1 | ∗ 𝜓 𝑗2 | ∗ · · · | ∗ 𝜓 𝑗𝑚 ∥𝑞 .

(1.19)

(1.20)

This is similar to working with a windowed scattering transform with a finite number of layers.

However, our operator is different from the operator 𝑆𝐽 in [11] because it does not contain the

12

filter 𝐴𝐽 to aggregate low frequency information, so the scale parameter in our formulation is not

bounded above or below. Additionally, because the averaging filter is replaced L𝑞 (R𝑛) norms, our

representation is fully translation invariant rather than translation invariant as 𝐽 → ∞.

As for the significance of using L𝑞 (R𝑛) norms to replace the averaging filter, there is one area

with direct application: quantum energy regression tasks [32], where a representation that is similar

to the rotation invariant representation in Section 6.2 has already been used for quantum energy

regression.

Given a configuration of atoms, we would like to estimate the ground state energy of the

configuration. Suppose we have a molecule with 𝐾 atoms with nuclear charges 𝑧𝑘 and nuclear

positions 𝑝𝑘 with 𝑘 = 1, . . . , 𝐾. The state 𝑥 of a molecule is given by

𝑥 = {( 𝑝𝑘 , 𝑧𝑘 ) ∈ R3 × R : 𝑘 = 1 . . . , 𝐾 },

(1.21)

Due to how we have defined our state, we would like our representation to have the following

properties:

• Permutation Invariance: the energy should not depend on the index of the molecules.

• Deformation Stability: small deformations of the molecule should only lead to small changes

in energy of the system.

• Isometry Invariance: the energy should be invariant to group actions such as translations,

rotations, and other general isometries.

• Multiscale Interactions: molecules have many interactions terms, and these interaction

terms depend on the pairwise distance between atoms (i.e. short range covalent bonds and

longer range Van Der Waals interactions).

The rotation invariant version of our scattering transform in Chapter 6 satisfies permutation

invariance, deformation stability, and has multiscale interactions based on the proofs we’ve provided.

We do not prove isometry invariance, but the operator is rotation and translation invariant.

Motivated by DFT theory, the paper [32] uses a dictionary of one and two layer scattering

norms with 𝑞 = 1 and 𝑞 = 2 to get (at the time) state-of-the-art results for energy regression tasks

for planar molecules.

In particular, scattering operators with 𝑞 = 1 scaled with the number of

13

atoms in the system and 𝑞 = 2 encoded pairwise interactions. The motivation for using 1 < 𝑞 < 2

comes from [33, 34], which based on the Thomas–Fermi–Dirac–von Weizsäcker model [35], also

use scattering norms with 𝑞 = 4/3, 5/3. Later papers, like [33, 34], use a similar representation,

involving spherical harmonics, for 3D quantum energy regression.

Remark 1. We can replace all the modulus operators with any contraction mapping (or use different

contraction mappings in each layer) in the definition above, and all the proofs in the rest of this paper

will still work. In particular, the modulus can be replaced with a complex version of the rectified
linear unit (ReLU) nonlinearity, max(0, Re(𝑎𝑖))𝑖=1,...,𝑛 for 𝑎 ∈ C𝑛, which is a popular choice for

complex neural networks. Nonetheless, we will use the modulus operator throughout this paper

without any loss of generality.

We provide a general roadmap for this chapter. First, we will cover notation, basic properties

about wavelets and the wavelet scattering operator, and harmonic analysis that will be necessary

for the paper. We then provide norms for an 𝑚-layer wavelet scattering transforms and prove that

the operators are well defined mappings into specific spaces when 1 ≤ 𝑞 ≤ 2. Next, we explore

conditions under which the 𝑚-layer scattering transform is stable to dilations, and we generalize

our results to diffeomorphisms. Lastly, in the last section of this chapter, we formulate two new

translation invariant operators that are stable to diffeomorphisms. The first is rotation equivariant,

and the second is rotation invariant. Our contributions include, but are not limited to, the following:

• We formulate an extension of the dyadic wavelet scattering operator for a finite, arbitrary

number of layers with parameter 𝑞 ∈ [1, 2] by applying L𝑞 (R𝑛) norms instead of L1(R𝑛)

norms. Additionally, we formulate a wavelet scattering operator with 𝑞 ∈ [1, 2] that uses a

continuous scale parameter, like the continuous wavelet transform.

• We create a new finite depth scattering norm using dyadic and continuous scales in the

case when 𝑞 ∈ [1, 2], and prove that the mappings are well defined and provide theoretical

justification for a broader class of wavelets that make the scattering transform Lipchitz

continuous to the action of 𝐶2 diffeomorphisms. However, the trade-off is that our stability

bound depends on the number of layers.

14

• We provide a condition for norm equivalence in the case of 𝑞 = 2 that is less stringent.

• In the case of 𝑞 ∈ (1, 2], we prove that our norm is stable to diffeomorphisms 𝜏 ∈ 𝐶2(R𝑛)

provided that ∥𝜏∥∞ < 1

2𝑛 and the wavelet and its first and second partial derivatives have

sufficient decay. In the case of 𝑞 = 1, we show stability to dilations.

• We extend our formulation to include invariance or equivariance to the action of rotations

𝑅 ∈ SO(𝑛).

15

CHAPTER 2

GENERALIZING THE NONWINDOWED SCATTERING TRANSFORM

The contents of this chapter were a joint work with Matthew Hirn and Anna Little. A journal

version of this chapter is published in [36]. We start by providing basic prerequisite knowledge that

will be necessary for the results in this chapter.

2.1 Fourier Transforms and Hardy Spaces

The Fourier transform of a function 𝑓 ∈ L1(R𝑛) is the function (cid:98)𝑓 ∈ L∞(R𝑛) defined as:

∀ 𝜔 ∈ R𝑛 ,

(cid:98)𝑓 (𝜔) :=

∫

R𝑛

𝑓 (𝑥)𝑒−𝑖𝑥·𝜔 𝑑𝑥 .

The Hilbert transform of a function 𝑓 ∈ L1(R) is denoted by 𝐻 𝑓 and is defined as:

𝐻 𝑓 (𝑥) := lim
𝜖→0

∫

|𝑥−𝑦|>𝜖

𝑓 (𝑦)
𝑥 − 𝑦

𝑑𝑦 .

The map 𝐻 is a convolution operator in which 𝑓 is convolved against the function 1/𝑥. We note

that

𝐻 : L𝑞 (R) → L𝑞 (R) , ∀ 1 < 𝑞 < ∞ ,

however the result is not true for 𝑞 = 1, i.e., if 𝑓 ∈ L1(R) it is not necessarily true that 𝐻 𝑓 ∈ L1(R).

We thus introduce the Hardy space. We denote the Hardy space as H1(R) and it consists of those

functions 𝑓 ∈ L1(R) such that 𝐻 𝑓 ∈ L1(R) as well. For 𝑓 ∈ H1(R) the Hardy space norm is

∥ 𝑓 ∥H1 (R), which we define as (see Corollary 2.4.7 of [37])

∥ 𝑓 ∥H1 (R) := ∥ 𝑓 ∥1 + ∥𝐻 𝑓 ∥1 .

(2.1)

One can show that if 𝑓 ∈ H1(R), then 𝑓 must necessarily have zero average. An important property

of the Hilbert transform and convolution is the following:

𝐻 ( 𝑓 ∗ 𝑔) = 𝐻 𝑓 ∗ 𝑔 = 𝑓 ∗ 𝐻𝑔 ,

𝑓 ∈ L𝑝 (R) , 𝑔 ∈ L𝑞 (R) ,

1 < 1
𝑝

+

1
𝑞

.

We have a similar definition for Hardy spaces when 𝑛 ≥ 2. For 1 ≤ 𝑗 ≤ 𝑛, define the 𝑗 th Riesz

transform as

𝑅 𝑗 𝑓 (𝑥) = lim
𝜀→0

∫

|𝑥−𝑦|>𝜀

𝑥 𝑗 − 𝑦 𝑗
|𝑥 − 𝑦|𝑛+1

𝑓 (𝑦) 𝑑𝑦 ,

(2.2)

16

where 𝑥 = (𝑥1, . . . , 𝑥𝑛) and 𝑦 = (𝑦1, . . . , 𝑦𝑛). The Hardy space 𝑓 ∈ H1(R𝑛) consists of functions 𝑓
such that 𝑓 ∈ L1(R𝑛) and 𝑅 𝑗 𝑓 ∈ L1(R𝑛) for 1 ≤ 𝑗 ≤ 𝑛 as well. For 𝑓 ∈ H1(R𝑛) the Hardy space

norm is ∥ 𝑓 ∥H1 (R𝑛), which we define as (see Corollary 2.4.7 of [37])

∥ 𝑓 ∥H1 (R𝑛) := ∥ 𝑓 ∥1 +

𝑛
∑︁

𝑗=1

∥𝑅 𝑗 𝑓 ∥1 .

(2.3)

2.1.1 Operator Valued Spaces

Consider a Banach space B. Suppose 𝑓

: R𝑛 → B and 𝑥 → ∥ 𝑓 (𝑥) ∥B is measurable in the

Lebesgue sense. Define L

𝑝

B (R𝑛) for 1 ≤ 𝑝 < ∞ to be

∥ 𝑓 ∥

𝑝

L

=

𝑝

B (R𝑛)

∫

R𝑛

∥ 𝑓 (𝑥) ∥

𝑝
B

𝑑𝑥 .

Also, for 1 ≤ 𝑝 < ∞, define

∥ 𝑓 ∥L

𝑝,∞

B (R𝑛) = sup
𝛿>0

𝛿 · 𝑚({𝑥 ∈ R𝑛 : ∥ 𝑓 (𝑥) ∥B > 𝛿})1/𝑝 .

We also have the following relation:

∥ 𝑓 ∥L

𝑝,∞

B (R𝑛) ≤ ∥ 𝑓 ∥L

𝑝

B (R𝑛) .

Note that for 𝑓 : R𝑛 → R𝑛,

∥ 𝑓 ∥

𝑝

𝑝
R𝑛 (R𝑛)

L

=

∫

R𝑛

∥ 𝑓 (𝑥) ∥

𝑝
R𝑛 𝑑𝑥 =

∫

R𝑛

| 𝑓 (𝑥)| 𝑝 𝑑𝑥 = ∥ 𝑓 ∥

𝑝
𝑝 .

2.2 Wavelet Scattering is a Bounded Operator

In this chapter we explore for which 𝑞 > 0 and 𝑚 ≥ 1 the wavelet scattering transforms 𝑆𝑚

cont,𝑞 𝑓
dyad,𝑞 𝑓 are well-defined as functions in some Banach space (i.e., have finite norm), and under

and 𝑆𝑚

what circumstances.

Let 𝜓 be a wavelet. We assume that 𝜓 has the following properties:

|𝜓(𝑥)| ≤ 𝐴(1 + |𝑥|)−𝑛−𝜀

∫

R𝑛

|𝜓(𝑥 − 𝑦) − 𝜓(𝑥)| 𝑑𝑥 ≤ 𝐴|𝑦|𝜀′ ,

17

(2.4)

(2.5)

for some constants 𝐴, 𝜀′, 𝜀 > 0 and for all ℎ ≠ 0.

Consider the Littlewood-Paley 𝐺-function

𝐺𝜓 ( 𝑓 )(𝑥) =

(cid:18)∫

(0,∞)

| 𝑓 ∗ 𝑡−𝑛𝜓(𝑥/𝑡)|2

(cid:19) 1/2

.

𝑑𝑡
𝑡

(2.6)

Let B = L2 (cid:16)
. We can rewrite this as a Bochner integral by considering the function
𝐾 (𝑥) = (𝑡−𝑛/2𝜓𝑡 (𝑥))𝑡>0. This is a mapping 𝐾 : R𝑛 → B and the function 𝑥 → ∥𝐾 (𝑥)∥B is

(0, ∞), 𝑑𝑡
𝑡

(cid:17)

measurable. Also, if we let

T ( 𝑓 )(𝑥) =

(cid:18)∫

R𝑛

𝑡−𝑛/2𝜓𝑡 (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦

(cid:19)

𝑡>0

(cid:16)

=

(𝑡−𝑛/2𝜓𝑡 ∗ 𝑓 ) (𝑥)

(cid:17)

,

𝑡>0

we observe that

and

𝐺𝜓 ( 𝑓 ) (𝑥) = ∥T ( 𝑓 ) (𝑥) ∥B

∥𝐺𝜓 ( 𝑓 ) ∥

𝑝
𝑝 = ∥T ( 𝑓 ) ∥

𝑝
𝐿 𝑝
B (R𝑛)

.

From Problem 6.1.4 of [38], the two properties above for the wavelet 𝜓 imply that

∥𝐾 (𝑥) ∥B ≤

𝑐𝑛 𝐴
|𝑥|𝑛

,

and

sup
𝑦∈R𝑛\{0}

∫

|𝑥|≥2|𝑦|

∥𝐾 (𝑥 − 𝑦) − 𝐾 (𝑥) ∥B 𝑑𝑥 ≤ 𝑐′

𝑛 𝐴 ,

(2.7)

(2.8)

where 𝑐𝑛 and 𝑐′

𝑛 depend only on 𝑛, 𝜀, and 𝜀′. We will omit the dependence on 𝜀 and 𝜀′ throughout

the rest of this paper, and this will have no effect on any of our proofs.

Remark 2. For the rest of this paper, we will write 𝐺 in place of 𝐺𝜓 when referring to the

𝐺-function because the dependence on the mother wavelet is clear.

Remark 3. Note that (2.5) holds under the alternative condition

|∇𝜓(𝑥)| ≤ 𝐴(1 + |𝑥|)−𝑛−1−𝜖 ′ .

(2.9)

This is a consequence of Mean Value Theorem.

18

We have the following result taken from Problem 6.1.4 of [38] and from Chapter V of [39].

Lemma 5 ([38, 39]). Assume that 𝜓 is defined as above and satisfies (2.7) and (2.8). Then the

operator 𝐺 is bounded from L2(R𝑛) to L2(R𝑛). Also, for 𝑝 ∈ (1, ∞) and B = L2(R+, 𝑑𝑡/𝑡), we

have

∥T 𝑓 ∥L

𝑝

B (R𝑛) ≤ 𝐶𝑛 𝐴 max( 𝑝, ( 𝑝 − 1)−1) ∥ 𝑓 ∥L 𝑝 (R𝑛) ,

for some 𝐶𝑛. For all 𝑓 ∈ L1(R𝑛), we also have

∥T 𝑓 ∥L1,∞

B (R𝑛) ≤ 𝐶′

𝑛 𝐴∥ 𝑓 ∥L1 (R𝑛)

and

for some 𝐶′
𝑛.

∥T 𝑓 ∥L1

B (R𝑛) ≤ 𝐶′

𝑛 𝐴∥ 𝑓 ∥H1 (R𝑛) ,

Remark 4. We can also formulate similar bounds for the Littlewood-Paley 𝔤 operator

𝔤( 𝑓 )(𝑥) :=

(cid:35) 1/2

|𝜓 𝑗 ∗ 𝑓 (𝑥)|2

(cid:34)

∑︁

𝑗 ∈Z

using similar arguments.

(2.10)

(2.11)

(2.12)

(2.13)

Remark 5. Let 𝜓 be a wavelet that has properties (2.4) and (2.5). Then with the L2 normalized

dilations, the Littlewood-Paley 𝐺-function can be written as:

𝐺 ( 𝑓 )(𝑥) =

(cid:20)∫ ∞

0

| 𝑓 ∗ 𝜓𝜆 (𝑥)|2

(cid:21) 1/2

.

𝑑𝜆
𝜆𝑛+1

(2.14)

Note that the 𝜆 measure for 𝐺 ( 𝑓 ) matches the measure in defining the norm of W 𝑓 .

2.2.1 The L2(R𝑛) Wavelet Scattering Transform

In this section we prove the L2(R𝑛) scattering transforms are bounded operators. More specif-

ically, we prove that 𝑆𝑚

cont,2 : L2(R𝑛) → L2(R𝑚

+ ), where L2(R𝑚

+ ) has the weighted measure defined

by

∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R𝑚

+ ) :=

∫ ∞

· · ·

∫ ∞

0

0

|𝑆𝑚

cont,2

𝑓 (𝜆1, . . . , 𝜆𝑚)|2

𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

19

and we show that ∥𝑆𝑚

cont,2

where

𝑓 ∥L2 (R𝑚

+ ) ≤ 𝐶 ∥ 𝑓 ∥L2 (R𝑛). We also show that 𝑆𝑚

dyad,2 : L2(R𝑛) → ℓ2(Z𝑚),

∥𝑆𝑚

dyad,2

𝑓 ∥2

ℓ2 (Z𝑚) :=

∑︁

𝑗𝑚∈Z

. . . ∑︁
𝑗1∈Z

|𝑆𝑚

dyad,2

𝑓 ( 𝑗1, . . . , 𝑗𝑚)|2.

Proposition 6. For any wavelet satisfying (2.4) and (2.5), we have 𝑆𝑚
dyad,2 : L2(R𝑛) → ℓ2(Z𝑚).
𝑆𝑚

cont,2 : L2(R𝑛) → L2(R𝑚

+ ) and

Proof. The proof of the dyadic case is essentially identical to the proof given below and is thus

omitted. The case of 𝑚 = 1 follows by an application of Fubini’s Theorem:

∥𝑆cont,2 𝑓 ∥2

L2 (R+) =

=

=

∫ ∞

0
∫ ∞

0
∫

R𝑛

∥ 𝑓 ∗ 𝜓𝜆 ∥2
2

𝑑𝜆
𝜆𝑛+1

∫

R𝑛

|( 𝑓 ∗ 𝜓𝜆) (𝑥)|2 𝑑𝑥

𝑑𝜆
𝜆𝑛+1

|𝐺 ( 𝑓 ) (𝑥)|2 𝑑𝑥

≤ 𝐶 ∥ 𝑓 ∥2
2

by boundedness of the G-function. Now we proceed by using induction. Assume that we have

∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R𝑚
+ )

≤ 𝐶𝑚 ∥ 𝑓 ∥2

2. Let W𝑡 𝑓 = 𝑓 ∗ 𝜓𝑡, define 𝑀 𝑓 = | 𝑓 |, and 𝑈𝜆 = 𝑀𝑊𝜆 for notational

brevity. Then notice that

∥||| 𝑓 ∗ 𝜓𝜆1 | ∗ 𝜓𝜆2 | ∗ · · · ∗ 𝜓𝜆𝑚 | ∗ 𝜓𝜆𝑚+1 ∥2

2 = ∥W𝜆𝑚+1

𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ∥2
2

.

Substituting yields

∥𝑆𝑚+1
cont,2

𝑓 ∥L2 (R𝑚+1

+

) =

=

=

∫ ∞

0
∫ ∞

0
∫ ∞

· · ·

· · ·

· · ·

∫ ∞

0
∫ ∞

0
∫ ∞

0

0

∥W𝜆𝑚+1

𝑈𝜆𝑚 · · · 𝑈𝜆1

∫ ∞

0

∥(𝑈𝜆𝑚 · · · 𝑈𝜆1

. . .

𝑓 ∥2
2

𝑑𝜆1
𝜆𝑛+1
1
𝑓 ) ∗ 𝜓𝜆𝑚+1 ∥2
2

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1
𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1

𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

∫ ∞

· · ·

≤ 𝐶

0
∫ ∞

· · ·

= 𝐶

∫ ∞

0
∫ ∞

0

0

≤ 𝐶𝑚+1∥ 𝑓 ∥2
2

,

𝑑𝜆1
𝜆𝑛+1
1
𝑑𝜆1
𝜆𝑛+1
1

. . .

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚
𝑑𝜆𝑚
𝜆𝑛+1
𝑚

𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

∥𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ∥2

L2 (R+)

∥𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ∥2
2

|𝑆𝑚

cont,2(𝜆1, . . . , 𝜆𝑚)|2

20

where we used the induction hypothesis in the last line. This completes the proof.

□

Proposition 7. Suppose 𝜓 is a Littlewood-Paley wavelet satisfying (2.4) and (2.5). Then 𝑆𝑚

𝑓 ∥1 = 𝐶𝑚

𝜓 ∥ 𝑓 ∥2

2. Also, 𝑆𝑚

cont,2

𝑓 :
dyad,2 : L2(R𝑛) → ℓ2(Z𝑚) and

cont,2

L2(R𝑛) → L2(R𝑚

∥𝑆𝑚

dyad,2

𝑓 ∥1 = ˆ𝐶𝑚

+ ) and specifically ∥𝑆𝑚
𝜓 ∥ 𝑓 ∥2
2.

Proof. We only provide the proof of the continuous case again. First consider the case 𝑚 = 1. We

have:

∥𝑆cont,2 𝑓 ∥2

L2 (R+) =

∫ ∞

0

∥ 𝑓 ∗ 𝜓𝜆 ∥2
2

𝑑𝜆
𝜆𝑛+1

∥ ˆ𝑓 · ˆ𝜓𝜆 ∥2
2

𝑑𝜆
𝜆𝑛+1

∫ ∞

0
∫ ∞

(cid:18)∫

=

=

=

=

=

1
(2𝜋)𝑛
1
(2𝜋)𝑛
1
(2𝜋)𝑛
1
(2𝜋)𝑛
1
(2𝜋)𝑛

0
∫

R𝑛
(cid:18)∫ ∞

0

R𝑛

∫

R𝑛

𝐶𝜓 ∥ ˆ𝑓 ∥2
2

| ˆ𝑓 (𝜔)|2| ˆ𝜓𝜆 (𝜔)|2 𝑑𝜔

(cid:19) 𝑑𝜆
𝜆𝑛+1

| ˆ𝜓(𝜆𝜔)|2

(cid:19)

𝑑𝜆
𝜆

| ˆ𝑓 (𝜔)|2 𝑑𝜔

(cid:16)

𝐶𝜓 | ˆ𝑓 (𝜔)|2(cid:17)

𝑑𝜔

= 𝐶𝜓 ∥ 𝑓 ∥2
2

.

Thus the claim holds for 𝑚 = 1. Now assume that it holds through 𝑚. Then by the inductive

hypothesis,

∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R+) =

∫ ∞

· · ·

∫ ∞

0

0

∥|| 𝑓 ∗ 𝜓𝜆1 | ∗ 𝜓𝜆2 | ∗ · · · ∗ 𝜓𝜆𝑚 ∥2
2

𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

= 𝐶𝑚

𝜓 ∥ 𝑓 ∥2
2

.

Now consider the case of 𝑚 + 1. Similar to the previous proposition, we have

∥𝑆𝑚+1
cont,2

𝑓 ∥2

L2 (R+) =

∫ ∞

· · ·

0

∫ ∞

= 𝐶𝜓

∫ ∞

(cid:32)∫ ∞

0
∫ ∞

0

· · ·

0

0

∥(𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1 ∥2
2

|𝑆𝑚

cont,2

𝑓 (𝜆1, . . . , 𝜆𝑚)|2

𝑑𝜆1
𝜆𝑛+1
1

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1
𝑑𝜆𝑚
𝜆𝑛+1
𝑚

. . .

(cid:33) 𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

= 𝐶𝜓 ∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R+)

= 𝐶𝑚+1
𝜓

∥ 𝑓 ∥2
2

.

Thus, the claim is proven by induction.

□

21

2.2.2 The L1(R𝑛) Wavelet Scattering Transform

Define the notation W𝑡 𝑓 = 𝑓 ∗ 𝜓𝑡, 𝑀 𝑓 = | 𝑓 |, and 𝑈𝑡 = 𝑀W𝑡. We now try to prove that for

𝑚 ∈ N, 𝑆𝑚

cont,1 : H1(R𝑛) → L2(R𝑚

+ ). The norm for 𝑆𝑚

cont,1

𝑓 is:

∥𝑆𝑚

cont,1

𝑓 ∥L2 (R𝑚

+ ) :=

=

(cid:32)∫ ∞

∫ ∞

∫ ∞

· · ·

0
(cid:32)∫ ∞

0

0

∫ ∞

∫ ∞

· · ·

0

0

0

|𝑆𝑚

cont,1

𝑓 (𝜆1, 𝜆2, . . . , 𝜆𝑚)|2

(cid:13)
(cid:13)(𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚

(cid:13)
(cid:13)

2
1

𝑑𝜆1
𝜆𝑛+1
1
𝑑𝜆1
𝜆𝑛+1
1

𝑑𝜆2
𝜆𝑛+1
2
𝑑𝜆2
𝜆𝑛+1
2

(cid:33) 1/2

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

(cid:33) 1/2

.

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

An analogous result will also hold for the operator H1(R𝑛) → ℓ2(Z𝑚

+ ) with norm

∥𝑆𝑚

dyad,1

𝑓 ∥ℓ2 (Z𝑚) :=

(cid:32)

∑︁

𝑗𝑚∈Z

. . . ∑︁
𝑗1∈Z

|𝑆𝑚

dyad,1

𝑓 ( 𝑗1, . . . , 𝑗𝑚)|2

(cid:33) 1/2

.

Before we begin, we will need an important multiplier property of the individual Riesz Trans-

forms:

𝜔 𝑗
|𝜔|
Let (cid:174)𝛼 = (𝛼1, . . . , 𝛼𝑛) be a multi-index with 𝑛-elements, and let 𝑡 = (𝑡1, . . . , 𝑡𝑛) ∈ R𝑛. We say

(cid:100)𝑅 𝑗 𝑓 (𝜔) = −𝑖

ˆ𝑓 (𝜔) .

(2.15)

that 𝜓 has 𝑘 vanishing moments if for all | (cid:174)𝛼| < 𝑘, we have

∫

R𝑛

(cid:0)Π𝑛
𝑖=1

𝑡𝛼𝑖
𝑖

(cid:1) 𝜓(𝑡)𝑑𝑡 = 0.

(2.16)

The following lemmas will be necessary.

Lemma 8 ([40]). Suppose that 𝜓 has 𝑁 vanishing moments, let 𝑀 > 1 be an integer, let (cid:174)𝛼 be

defined as before, and let (cid:174)𝛽 = (𝛽1, . . . , 𝛽𝑛) be a multi-index. Assume that 𝜓 satisfies the following

properties:

• 𝜓 ∈ H𝑠 (R𝑑) ∩ 𝐶 (R𝑑) for some 𝑠 > 𝑀 + 𝑛
2 .

• There exists 𝐴 > 0 and 𝜖 ∈ [0, 1) such that 𝜓 satisfies

|𝐷 (cid:174)𝛼𝜓| ≤ 𝐴(1 + |𝑥|)−𝑛−𝑁−| (cid:174)𝛼|+𝜀 for 0 ≤ | (cid:174)𝛼| ≤ 𝑀.

• For 0 ≤ | (cid:174)𝛼| ≤ 𝑀 − 1 and | (cid:174)𝛽| < 𝑁 + | (cid:174)𝛼|,

∫

R𝑛

Π𝑛
𝑖=1

𝑡 𝛽𝑖
𝑖 𝐷 (cid:174)𝛼𝜓(𝑡) 𝑑𝑡 = 0.

22

Then

|𝐷 (cid:174)𝛼 𝑅𝑖𝜓(𝑥)| = |𝑅𝑖 𝐷 (cid:174)𝛼𝜓(𝑥)| ≤ 𝐴(1 + |𝑥|)−𝑛−𝑁−| (cid:174)𝛼|+𝜀+𝛿

for some 0 < 𝛿 < 1 − 𝜀 and 𝐷 (cid:174)𝛼 𝑅𝑖𝜓 has vanishing moments up to degree 𝑁 − 1 + | (cid:174)𝛼|.

An immediate consequence is the following Lemma, which we will provide without proof.

Lemma 9. Suppose that 𝜓 satisfies the following conditions:

• 𝜓 ∈ H𝑠 (R𝑑) ∩ 𝐶 (R𝑑) for some 𝑠 > 2 + 𝑛
2 .

• There exists 𝐴 > 0 and 𝜖 ∈ [0, 1) such that 𝜓 satisfies

|𝐷 (cid:174)𝛼𝜓| ≤ 𝐴(1 + |𝑥|)−𝑛−2−| (cid:174)𝛼|+𝜀 for 0 ≤ | (cid:174)𝛼| ≤ 3.

• For 0 ≤ | (cid:174)𝛼| ≤ 2 and | (cid:174)𝛽| < 2 + | (cid:174)𝛼|,

∫

R𝑛

Π𝑛
𝑖=1

𝑡 𝛽𝑖
𝑖 𝐷 (cid:174)𝛼𝜓(𝑡) 𝑑𝑡 = 0.

Then 𝑅 𝑗 𝜓 and all of its first and second partial derivatives have 𝑂 ((1 + |𝑥|)−𝑛−1+𝜂) decay for some

𝜂 ∈ (0, 1).

The first implication to take note of is that 𝑅 𝑗 𝜓 is a wavelet with "good" decay of itself and

all its first and second partial derivatives. Note that the strict decay on the partial derivatives is

necessary for technical reasons in later proofs, but decay on all second partial derivatives can be

relaxed for the following theorem.

Theorem 10. Let 𝜓 be a wavelet satisfying Lemma 9 and let 𝑆𝑚

cont,1 be defined as above. Then for

𝑓 ∈ H1(R𝑛), there exists a constant 𝐶𝑚 such that

Additionally,

∥𝑆𝑚

cont,1

𝑓 ∥L2 (R𝑚

+ ) ≤ 𝐶𝑚 ∥ 𝑓 ∥H1 (R𝑛) .

∥𝑆𝑚

dyad,1

𝑓 ∥ℓ2 (Z𝑚) ≤ 𝐶𝑚 ∥ 𝑓 ∥H1 (R𝑛).

23

Proof. We proceed by induction and only provide a proof for the continuous case because the

dyadic case follows by almost identical reasoning. Let 𝑓 ∈ H1(R𝑛) throughout the proof. By

Minkowski’s integral inequality ([41], Theorem 202), we have

∥ 𝑓 ∗ 𝜓𝜆 ∥2
1

(cid:19) 1/2

𝑑𝜆
𝜆𝑛+1

(cid:18)∫ ∞

0
(cid:32)∫ ∞

(cid:18)∫

0
(cid:32)∫

R𝑛

(cid:18)∫ ∞

R𝑛

0

∥𝑆cont,1 𝑓 ∥L2 (R+) =

=

≤

=

∫

R𝑛

𝐺 ( 𝑓 ) (𝑥) 𝑑𝑥

| 𝑓 ∗ 𝜓𝜆 (𝑥)| 𝑑𝑥

(cid:33) 1/2

(cid:19) 2 𝑑𝜆
𝜆𝑛+1

| 𝑓 ∗ 𝜓𝜆 (𝑥)|2

(cid:19) 1/2

(cid:33)

𝑑𝑥

𝑑𝜆
𝜆𝑛+1

= ∥𝐺 ( 𝑓 ) ∥1

≤ 𝐶 ∥ 𝑓 ∥H1 (R𝑛) ,

where in the last inequality we used Lemma 5.

Now we assume that there exists some 𝑚 ≥ 1 such that

∥𝑆𝑚

cont,1

𝑓 ∥L2 (R𝑚

+ ) ≤ 𝐶𝑚 ∥ 𝑓 ∥H1 (R𝑛).

24

We have

𝑓 ∥L2 (R𝑚+1

)

+

∥𝑆𝑚+1
cont,1
(cid:32)∫ ∞

=

0
(cid:32)∫ ∞

0

=

∫ ∞

0

≤ (cid:169)
(cid:173)
(cid:173)
(cid:171)
(cid:32)∫ ∞

=

0
(cid:32)∫ ∞

0
(cid:32)∫ ∞

=

=

· · ·

· · ·

∫ ∞

0

(cid:13)
(cid:13)(𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1

(cid:13)
2
(cid:13)
1

𝑑𝜆1
𝜆𝑛+1
1

∫ ∞

(cid:18)∫

0

R𝑛

(cid:12)
(cid:12)(𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1

(cid:12)
(cid:12) 𝑑𝑥

(cid:33) 1/2

· · ·

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1
(cid:19) 2 𝑑𝜆1
𝜆𝑛+1
1

· · ·

(cid:34)∫ ∞

R𝑛

0

(cid:12)
(cid:12)(𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1

2 𝑑𝜆𝑚+1
(cid:12)
(cid:12)
𝜆𝑛+1
𝑚+1

𝐺 (𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ) (𝑥) 𝑑𝑥

0

R𝑛

∥𝐺 (𝑈𝜆𝑚 · · · 𝑈𝜆1

𝑓 ) ∥2
1

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

(cid:33) 1/2

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

· · ·

(cid:21) 2 𝑑𝜆1
𝜆𝑛+1
1
(cid:33) 1/2

∫

(cid:169)
(cid:173)
(cid:171)
(cid:20)∫

· · ·

∫ ∞

0

∫ ∞

∫ ∞

0

∫ ∞

· · ·

· · ·

· · ·

∥𝐺 (W𝜆𝑚𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ) ∥2
1

(cid:33) 1/2

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

0

0

(cid:33) 1/2

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1

(cid:35) 1/2

2

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

1/2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

𝑑𝑥(cid:170)
(cid:174)
(cid:172)

since the 𝐺 function has a modulus already.

It follows that

∥𝑆𝑚

cont,1

𝑓 ∥L2 (R𝑚

+ ) ≤ 𝐶

(cid:32)∫ ∞

∫ ∞

· · ·

0

0

∥W𝜆𝑚𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ∥2

H1 (R𝑛)

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

(cid:33) 1/2

.

Now use the definition of the H1(R𝑛) norm to write

∥W𝜆𝑚𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ∥H1 (R𝑛) = ∥W𝜆𝑚𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ∥L1 (R𝑛)

+

𝑛
∑︁

𝑗=1

(cid:13)
(cid:13)

(cid:0)𝑅 𝑗W𝜆𝑚

(cid:1) (𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 )(cid:13)

(cid:13)L1 (R𝑛)

.

Thus, since 𝑅 𝑗W𝜆𝑚 ℎ = ℎ ∗ (cid:0)𝑅 𝑗 𝜓𝜆𝑚

(cid:1) and 𝑅 𝑗 𝜓 wavelet, we can use our induction hypothesis and the

25

previous lemma to get

(cid:32)∫ ∞

𝐶

∫ ∞

· · ·

∥W𝜆𝑚 (𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ) ∥2

H1 (R𝑛)

(cid:33) 1/2

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

∥W𝜆𝑚 (𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ) ∥2

L1 (R𝑛)

𝑑𝜆1
𝜆𝑛+1
1

· · ·

∫ ∞

0

(cid:13)
(cid:13)

(cid:0)𝑅 𝑗W𝜆𝑚

(cid:1) (𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 )(cid:13)
2
(cid:13)
L1 (R𝑛)

(cid:33) 1/2

(cid:33) 1/2

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

𝑑𝜆1
𝜆𝑛+1
1

0
(cid:32)∫ ∞

0

· · ·

≤ 𝐶

∫ ∞

0
(cid:32)∫ ∞

𝑛
∑︁

0

· · ·

𝑗=1

0

+ 𝐶

≤ 𝐶𝑚+1∥ 𝑓 ∥H1 (R𝑛).

Thus, the theorem is proved by induction.

□

The case of 𝑛 = 1 is a little trickier. We have the following multiplier property for the Hilbert

Transform:

(cid:99)𝐻 𝑓 (𝜔) =

+𝑖 (cid:98)𝑓 (𝜔) 𝜔 < 0
−𝑖 (cid:98)𝑓 (𝜔) 𝜔 > 0





(2.17)

Unfortunately, this yields less regularity for (cid:99)𝐻 𝑓 at the origin without additional assumptions.

However, notice that the Hilbert transform commutes with dilations, so in particular:

𝐻 (𝜓𝜆) = 𝐻 (𝜓)𝜆

and 𝐻 (𝜓 𝑗 ) = 𝐻 (𝜓) 𝑗 .

Using the calculation of (cid:99)𝐻 𝑓 in (2.17) we see that

𝐻𝜓 = −𝑖𝜓 ,

if 𝜓 is complex analytic.

Thus, we have the following corollary.

Corollary 11. Let 𝜓 be a complex analytic wavelet such that (2.4) and (2.5) hold. Then for

𝑓 ∈ H1(R), there exists a constant 𝐶𝑚 such that

Additionally,

∥𝑆𝑚

cont,1

𝑓 ∥L2 (R𝑚

+ ) ≤ 𝐶𝑚 ∥ 𝑓 ∥H1 (R) .

∥𝑆𝑚

dyad,1

𝑓 ∥ℓ2 (Z𝑚) ≤ 𝐶𝑚 ∥ 𝑓 ∥H1 (R).

26

2.2.3 L𝑞 (R𝑛) Wavelet Scattering Transform

In this section, assume 1 < 𝑞 < 2. We prove that for 𝑚 ∈ N, 𝑆𝑚

cont,𝑞 : L𝑞 (R𝑛) → L2(R𝑚

+ ). The

norm for 𝑆𝑚

cont,𝑞 𝑓 is:

∥𝑆𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚

+ ) :=

=

(cid:32)∫ ∞

∫ ∞

∫ ∞

· · ·

0
(cid:32)∫ ∞

0

0

∫ ∞

∫ ∞

· · ·

0

0

0

|𝑆𝑚

cont,𝑞 𝑓 (𝜆1, 𝜆2, . . . , 𝜆𝑚)|2

(cid:16)(cid:13)
(cid:13)(𝑈𝜆𝑚−1 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚

(cid:13)
(cid:13)𝑞

There is also an analagous result for

𝑑𝜆1
𝑑𝜆2
𝜆𝑛+1
𝜆𝑛+1
1
2
(cid:17) 2 𝑑𝜆1
𝜆𝑛+1
1

𝑑𝜆2
𝜆𝑛+1
2

(cid:33) 𝑞/2

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

(cid:33) 𝑞/2

.

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

∥𝑆𝑚

dyad,𝑞 𝑓 ∥

𝑞
ℓ2 (Z𝑚) :=

(cid:32)

∑︁

∑︁

· · ·

𝑗𝑚∈Z

𝑗𝑚∈Z

|𝑆𝑚

dyad,𝑞 𝑓 (𝜆1, 𝜆2, . . . , 𝜆𝑚)|2

(cid:33) 𝑞/2

.

Theorem 12. Let 1 < 𝑞 < 2. Also, let 𝜓 be a wavelet that satisfies properties (2.4) and (2.5) and

let 𝑆𝑚

cont,𝑞 and 𝑆𝑚
cont,𝑞 𝑓 ∥

𝑞
L2 (R+)

∥𝑆𝑚

dyad,𝑞 be defined as above. Then there exists a universal constant 𝐶𝑚 > 0 such that
≤ 𝐶𝑚 ∥ 𝑓 ∥

𝑞
𝑞 for all 𝑓 ∈ L𝑞 (R𝑛), and furthermore ∥𝑆𝑚

≤ 𝐶𝑚 ∥ 𝑓 ∥

𝑞
𝑞.

dyad,𝑞 𝑓 ∥

𝑞
ℓ2 (Z)

Proof. We proceed by induction and consider the case of 𝑚 = 1 first. Let 𝑓 ∈ L𝑞 (R𝑛). For the

continuous wavelet transform, we apply Minkowski’s integral inequality:

(cid:20)∫ ∞

0
(cid:34)∫ ∞

(cid:18)∫

0

R𝑛

∥𝑆cont,𝑞 𝑓 ∥

𝑞
L2 (R+)

=

=

≤

(cid:0)∥ 𝑓 ∗ 𝜓𝜆 ∥𝑞(cid:1) 𝑞 𝑑𝜆
𝜆𝑛+1

(cid:21) 1/2

| 𝑓 ∗ 𝜓𝜆 (𝑥)|𝑞 𝑑𝑥

(cid:35) 𝑞/2

(cid:19) 2/𝑞 𝑑𝜆
𝜆𝑛+1

∫

(cid:18)∫ ∞

R𝑛

0

| 𝑓 ∗ 𝜓𝜆 (𝑥)|2

(cid:19) 𝑞/2

𝑑𝑥

𝑑𝜆
𝜆𝑛+1

= ∥𝐺 ( 𝑓 ) ∥

𝑞
𝑞

≤ 𝐶 ∥ 𝑓 ∥

𝑞
𝑞.

where in the last inequality we used Theorem 5.

Now, let us assume that

∥𝑆𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚
+ )

≤ 𝐶𝑚·𝑞 ∥ 𝑓 ∥

𝑞
L𝑞 (R𝑛)

.

27

We apply Minkowski’s Integral inequality [41] to swap and then bound:

∥𝑆𝑚+1

cont,𝑞 𝑓 ∥
(cid:34)∫ ∞

𝑞
L2 (R𝑚+1
+
∫ ∞

· · ·

0
(cid:34)∫ ∞

· · ·

0

∫ ∞

(cid:18)∫

0

0

R𝑛

)

(cid:16)(cid:13)
(cid:13)(𝑈𝜆1 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1

(cid:13)
(cid:13)𝑞

(cid:17) 2/𝑞 𝑑𝜆1
𝜆𝑛+1
1

|(𝑈𝜆1 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1 (𝑥)|𝑞 𝑑𝑥

(cid:35) 𝑞/2

. . .

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1
(cid:19) 2/𝑞 𝑑𝜆1
𝜆𝑛+1
1

· · ·

∫ ∞

(cid:34)∫ ∞

(cid:18)∫

0

0

R𝑛

|(𝑈𝜆1 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1 (𝑥)|𝑞 𝑑𝑥

· · ·

∫ ∞

0

∫

(cid:32)∫ ∞

R𝑛

0








|(𝑈𝜆1 · · · 𝑈𝜆1

𝑓 ) ∗ 𝜓𝜆𝑚+1 (𝑥)|2

∫ ∞

· · ·

0

0

∥𝐺 (𝑈𝜆1 · · · 𝑈𝜆1

𝑓 )∥2
𝑞

. . .

(cid:35) 𝑞/2

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

≤ 𝐶𝑞

(cid:34)∫ ∞

∫ ∞

· · ·

0

0

∥(𝑈𝜆1 · · · 𝑈𝜆1) 𝑓 ∥2
𝑞

𝑑𝜆1
𝜆𝑛+1
1
𝑑𝜆1
𝜆𝑛+1
1

=

=

=

≤

=

0

∫ ∞














(cid:34)∫ ∞

∫ ∞

0

(cid:35) 𝑞/2

. . .

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1
𝑞
2 · 2
(cid:19) 2/𝑞 𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1

(cid:35)

𝑞 𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑞/2

𝑑𝜆𝑚
𝜆𝑛+1
𝑚







𝑞/2

(cid:33) 𝑞/2

𝑑𝜆𝑚+1
𝜆𝑛+1
𝑚+1

𝑑𝑥

2
𝑞








𝑑𝜆1
𝜆𝑛+1
1

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚









(cid:35) 𝑞/2

. . .

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

□

= 𝐶𝑞 ∥𝑆𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚
+ )

≤ 𝐶 (𝑚+1)𝑞 ∥ 𝑓 ∥

𝑞
𝑞.

This proves the desired claim.

2.3 Stability to Dilations

We now consider dilations defined by 𝜏(𝑥) = 𝑐𝑥 for some constant 𝑐, so that 𝐿𝜏 𝑓 (𝑥) =

𝑓 ((1 − 𝑐)𝑥). We will start by proving a lemma that will be useful for our work.

Lemma 13. Assume 𝐿𝜏 is defined as above. Then

𝐿𝜏 𝑓 ∗ 𝜓𝜆 (𝑥) = (1 − 𝑐)−𝑛/2 (cid:0) 𝑓 ∗ 𝜓(1−𝑐)𝜆(cid:1) ((1 − 𝑐)𝑥).

Proof. Notice that

𝐿𝜏 𝑓 ∗ 𝜓𝜆 (𝑥) =

∫

R𝑛

𝑓 ((1 − 𝑐)𝑦)𝜓𝜆 (𝑥 − 𝑦) 𝑑𝑦.

28

We make the substitution 𝑧 = (1 − 𝑐)𝑦. Then it follows that

𝐿𝜏 𝑓 ∗ 𝜓𝜆 (𝑥) = (1 − 𝑐)−𝑛 ∫
= (1 − 𝑐)−𝑛 ∫

𝑓 (𝑧)𝜓𝜆 (𝑥 − (1 − 𝑐)−1𝑧) 𝑑𝑧

𝑓 (𝑧)𝜆−𝑛/2𝜓

(cid:16)

𝜆−1(𝑥 − (1 − 𝑐)−1𝑧)

(cid:17)

𝑑𝑧

R𝑛

R𝑛
∫

R𝑛

∫

= (1 − 𝑐)−𝑛/2

𝑓 (𝑧) [(1 − 𝑐)𝜆]−𝑛/2𝜓

(cid:16)

[(1 − 𝑐)𝜆]−1 ((1 − 𝑐)𝑥 − 𝑧)

(cid:17)

𝑑𝑧

= (1 − 𝑐)−𝑛/2

𝑓 (𝑧)𝜓(1−𝑐)𝜆 ((1 − 𝑐)𝑥 − 𝑧) 𝑑𝑧

R𝑛
= (1 − 𝑐)−𝑛/2 𝑓 ∗ 𝜓(1−𝑐)𝜆 ((1 − 𝑐)𝑥)

= (1 − 𝑐)−𝑛/2𝐿𝜏 (cid:0) 𝑓 ∗ 𝜓(1−𝑐)𝜆(cid:1) (𝑥).

□

Remark 6. We also have

𝐿𝜏W𝜆 𝑓 (𝑥) = ( 𝑓 ∗ 𝜓𝜆) (𝑥(1 − 𝑐)).

Before we begin the next Lemma, we explain the general idea behind our approach to explain

the necessity of Lemma 14. Define

Ψ(𝑥) = (1 − 𝑐)−𝑛/2𝜓(1−𝑐) (𝑥) − 𝜓(𝑥).

(2.18)

We want to prove that Ψ satisfies (2.4) and (2.5) with a linear dependence on 𝑐 for future stability

lemmas.

Lemma 14. Suppose that 𝜓 is a wavelet that satisfies the following three conditions:

|𝜓(𝑥)| ≤

|∇𝜓(𝑥)| ≤

∥𝐷2𝜓(𝑥)∥∞ ≤

𝐴
(1 + |𝑥|)𝑛+1+𝛼
𝐴
(1 + |𝑥|)𝑛+1+𝛽
𝐴
(1 + |𝑥|)𝑛+1+𝜅

𝑥 ∈ R𝑛,

𝑥 ∈ R𝑛,

𝑥 ∈ R𝑛,

Ψ(𝑥) = (1 − 𝑐)−𝑛/2𝜓(1−𝑐) (𝑥) − 𝜓(𝑥).

(2.19)

(2.20)

(2.21)

for 𝛼, 𝛽, 𝜅 > 0. Consider

for 𝑐 < 1

2𝑛 . Then Ψ is a wavelet satisfying (2.4) and (2.5).

29

Proof. Without loss of generality, assume 𝛼 < 𝛽 < 𝜅 < 1. First, it’s clear that ∫

R𝑛 Ψ = 0. We now
just need to verify properties (2.4) and (2.5). Assume 𝑐 > 0. We can modify the proof accordingly

if 𝑐 < 0. Then

|Ψ(𝑥)| =

(cid:12)
(cid:12)
(cid:12)

(1 − 𝑐)−𝑛/2𝜓(1−𝑐) (𝑥) − 𝜓(𝑥)
(cid:18)

(cid:19)

(cid:12)
(cid:12)
(cid:12)

= (1 − 𝑐)−𝑛

𝜓

− (1 − 𝑐)𝑛𝜓 (𝑥)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑥
(1 − 𝑐)
𝑥
(cid:17)
1 − 𝑐

≤ (1 − 𝑐)−𝑛

(cid:16)

𝜓

− 𝜓

(cid:18) 1 − 𝑐
1 − 𝑐

𝑥

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

+ (1 − 𝑐)−𝑛

𝑛
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛
𝑗

𝑐 𝑗 |𝜓 (𝑥)| .

Now use mean value theorem on the first term to choose a point 𝑧 on the segment connecting 𝑥
1−𝑐

and 𝑥 such that

𝑐
1 − 𝑐

(cid:12)[∇𝜓(𝑧)]𝑇 𝑥(cid:12)
(cid:12)

(cid:12) =

𝜓

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:17)

(cid:16)

𝑥
1 − 𝑐

− 𝜓

(cid:18) 1 − 𝑐
1 − 𝑐

𝑥

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

.

We now use Cauchy-Schwarz to bound the left side:

𝑐
1 − 𝑐

(cid:12)[∇𝜓(𝑧)]𝑇 𝑥(cid:12)
(cid:12)

(cid:12) ≤

𝑐
1 − 𝑐

𝐴|𝑥|
(1 + |𝑧|)𝑛+1+𝛽

.

Since 𝑧 lies on the segment connecting 𝑥

1−𝑐 and 𝑥, we see that for some 𝑡 ∈ [0, 1], we have

+ 𝑡𝑥

𝑥

=

𝑥
1 − 𝑐
𝑥 +

𝑧 = (1 − 𝑡)
1 − 𝑡
1 − 𝑐
1 − 𝑡 + 𝑡 − 𝑡𝑐
1 − 𝑐
𝑥.

𝑡 − 𝑡𝑐
1 − 𝑐
𝑥

=

=

1 − 𝑡𝑐
1 − 𝑐

Thus, |𝑧| ≥ |𝑥|. It now follows that

𝑐
1 − 𝑐

𝐴|𝑥|
(1 + |𝑧|)𝑛+1+𝛽 ≤

𝑐
1 − 𝑐

𝐴
(1 + |𝑥|)𝑛+𝛽

.

Finally, we get

|Ψ𝜆 (𝑥)| ≤

(cid:205)𝑛

(cid:1)𝑐 𝑗

(cid:0)𝑛
𝑗

𝑗=1

𝐴
(1 + |𝑥|)𝑛+𝛽 +
(1 − 𝑐)𝑛+1
(cid:0)𝑛
(cid:1)𝑐 𝑗
(cid:19) −𝑛−1 (cid:205)𝑛
𝑗=1
𝑗
(1 + |𝑥|)𝑛+𝛼

𝐴
(1 + |𝑥|)𝑛+𝛼

𝑐
(1 − 𝑐)𝑛+1
(cid:18) 2𝑛
2𝑛 − 1
𝐴𝑛𝑐
(1 + |𝑥|)𝑛+𝛼

≤ 2𝐴

≤

30

for some constant 𝐴𝑛 since we assume 𝛼 < 𝛽 and 𝑐 < 1

2𝑛 . Thus, (2.4) is satisfied.

We use a similar idea for proving (2.5) holds. Assume 𝑐 > 0 without loss of generality and

further assume that |𝑥| ≥ 2|𝑦|. By Mean Value Theorem, there exists 𝑧 on the line segment

connecting 𝑥 and 𝑥 − 𝑦 such that

Like before, we notice that

|Ψ(𝑥 − 𝑦) − Ψ(𝑥)| = |∇Ψ(𝑧)||𝑦|.

|∇Ψ(𝑧)| =

(1 − 𝑐)−𝑛/2∇𝜓(1−𝑐) (𝑧) − ∇𝜓(𝑧)
(cid:16)

(cid:17)

(1 − 𝑐)−𝑛−1∇𝜓

(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
=
(cid:12)
= (1 − 𝑐)−𝑛−1 (cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤ (1 − 𝑐)−𝑛−1

𝑧
1 − 𝑐
𝑧
(cid:16)
1 − 𝑐
𝑧
1 − 𝑐

(cid:16)

(cid:17)

(cid:17)

∇𝜓

∇𝜓

− ∇𝜓(𝑧)

(cid:12)
(cid:12)
(cid:12)
− (1 − 𝑐)𝑛+1∇𝜓(𝑧)

(cid:12)
(cid:12)
(cid:12)

− ∇𝜓

(cid:18) 1 − 𝑐
1 − 𝑐

𝑧

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

+ (1 − 𝑐)−𝑛−1

𝑛+1
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛 + 1
𝑗

𝑐 𝑗 |∇𝜓 (𝑧)| .

Let 𝑆 be the set of points on the segment connecting 𝑧

1−𝑐 and 𝑧. By Mean Value Inequality, since 𝑆

is closed and bounded, we have

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∇𝜓

(cid:17)

(cid:16)

𝑧
1 − 𝑐

− ∇𝜓

(cid:18) 1 − 𝑐
1 − 𝑐

𝑧

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤

𝑐
1 − 𝑐 max

𝑤∈𝑆

(cid:13)𝐷2𝜓(𝑤)(cid:13)
(cid:13)

(cid:13)∞ |𝑧|.

The maximum for the quantity above is attained in 𝑆, so let us say the maximizer is 𝑤1 = (1−𝑡) 𝑧
for some 𝑡 ∈ [0, 1]. Now use decay of the Hessian to bound the right side:

1−𝑐 +𝑡𝑧

𝑐
1 − 𝑐 max

𝑤∈𝑆

(cid:13)𝐷2𝜓(𝑤)(cid:13)
(cid:13)

(cid:13)∞ |𝑧| ≤

𝑐
1 − 𝑐

𝐴|𝑧|
(1 + |𝑤1|)𝑛+1+𝜅

.

It follows that

+ 𝑡𝑧

𝑧

=

𝑧
1 − 𝑐
𝑧 +

𝑤1 = (1 − 𝑡)
1 − 𝑡
1 − 𝑐
1 − 𝑡 + 𝑡 − 𝑡𝑐
1 − 𝑐
𝑧.

𝑡 − 𝑡𝑐
1 − 𝑐
𝑧

=

=

1 − 𝑡𝑐
1 − 𝑐

31

Thus, |𝑤1| ≥ |𝑧|. We conclude
𝑐
1 − 𝑐

For bounding |∇Ψ(𝑧)|, we see

𝐴|𝑧|
(1 + |𝑤1|)𝑛+1+𝜅 ≤

𝑐
1 − 𝑐

𝐴
(1 + |𝑧|)𝑛+𝜅 .

(cid:1)𝑐 𝑗

(cid:0)𝑛+1
𝑗

(cid:205)𝑛+1
𝑗=1
(1 − 𝑐)𝑛+1

𝐴
(1 + |𝑧|)𝑛+1+𝛽

|∇Ψ(𝑧)| ≤

𝑐
(1 − 𝑐)𝑛+2

𝐴
(1 + |𝑧|)𝑛+𝜅 +
(cid:0)𝑛+1
(cid:1)𝑐 𝑗
2 (cid:205)𝑛+1
𝑗=1
𝑗
(1 + |𝑧|)𝑛+𝜅
(cid:0)𝑛+1
(cid:19) 𝑛+2 2𝐴 (cid:205)𝑛+1
𝑗=1
𝑗
(1 + |𝑧|)𝑛+𝜅

(cid:1)𝑐 𝑗

.

≤ 𝐴(1 − 𝑐)−𝑛−2

≤

(cid:18) 2𝑛
2𝑛 − 1

Going back to proving (2.5) holds for Ψ,

|Ψ(𝑥 − 𝑦) − Ψ(𝑥)| = |∇Ψ(𝑧)||𝑦| ≤

(cid:18) 2𝑛
2𝑛 − 1

(cid:19) 𝑛+2 2𝐴 (cid:205)𝑛+1
𝑗=1

(cid:1)𝑐 𝑗 |𝑦|

(cid:0)𝑛+1
𝑗
(1 + |𝑧|)𝑛+𝜅

.

since the point 𝑧 lies on the lines on a line segment connecting 𝑥 − 𝑦 and 𝑥 with |𝑥| ≥ 2|𝑦|, we can

use an argument similar to above to conclude

|Ψ(𝑥 − 𝑦) − Ψ(𝑥)| ≤ 2𝑛+1+𝜅

(cid:18) 2𝑛
2𝑛 − 1

(cid:0)𝑛+1
(cid:1)𝑐 𝑗
(cid:19) 𝑛+2 𝐴 (cid:205)𝑛+1
𝑗=1
𝑗
(1 + |𝑥|)𝑛+𝜅

|𝑦|.

Now integrate to get

∫

|𝑥|≥2|𝑦|

|Ψ(𝑥 − 𝑦) − Ψ(𝑥)| 𝑑𝑥 ≤ 2𝑛+1+𝜅

= 2𝑛+1+𝜅

(cid:18) 2𝑛
2𝑛 − 1

(cid:18) 2𝑛
2𝑛 − 1

(cid:19) 𝑛+2

𝐴

𝑛+1
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛 + 1
𝑗

∫

𝑐 𝑗 |𝑦|

|𝑥|≥2|𝑦|

𝑑𝑥
|𝑥|𝑛+𝜅

(cid:19) 𝑛+2

𝐴𝐼𝑛

𝑛+1
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛 + 1
𝑗

𝑐 𝑗 |𝑦|1−𝜅,

where 𝐼𝑛 is some constant associated with the integration. Finally, we have a bound of

∫

|𝑥|≥2|𝑦|

|Ψ(𝑥 − 𝑦) − Ψ(𝑥)| 𝑑𝑥 ≤ ˜𝐴𝑛𝑐|𝑦|1−𝜅.

for some constant

˜𝐴𝑛 only dependent on the dimension 𝑛. Thus, (2.5) holds with exponent

1 − 𝜅 ∈ (0, 1). Let ˆ𝐴𝑛 = max{𝐴𝑛, ˜𝐴𝑛}. It follows that

ˆ𝐴𝑛𝑐
(1 + |𝑥|)𝑛+𝛼

|Ψ𝜆 (𝑥)| ≤

∫

|𝑥|≥2|𝑦|

|Ψ(𝑥 − 𝑦) − Ψ(𝑥)| 𝑑𝑥 ≤ ˆ𝐴𝑛𝑐|𝑦|1−𝜅.

□

32

It follows from Problem 6.1.2 in [38] that the bound in the 𝐺-function depends linearly on the

constant 𝐴 from Theorem 5 when proving L2(R𝑛) boundedness. Thus, the following corollaries

hold.

Corollary 15. Assume |𝑐| < 1

2𝑛 . For 𝜓 satisfying the conditions of Lemma 14, when 1 < 𝑝 < ∞,

there exist constants 𝐶𝑛,𝑝 and ˆ𝐶𝑛,𝑝 such that

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

(cid:18)∫ ∞

0

| 𝑓 ∗ Ψ𝜆 (𝑥)|2

𝑑𝜆
𝜆𝑛+1

(cid:19) 1/2(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)L 𝑝 (R𝑛)

≤ 𝑐 · 𝐶𝑛,𝑝 max{𝑝, ( 𝑝 − 1)−1}∥ 𝑓 ∥L 𝑝 (R𝑛)

and

(cid:32)

∑︁

(cid:33) 1/2(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)L 𝑝 (R𝑛)
Alternatively, if one of the following holds:

| 𝑓 ∗ Ψ𝑗 (𝑥)|2

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑗 ∈Z

≤ 𝑐 · ˆ𝐶𝑛 max{𝑝, ( 𝑝 − 1)−1}∥ 𝑓 ∥L 𝑝 (R𝑛).

• 𝑛 = 1, 𝜓 is complex analytic and satisfies the conditions of Lemma 14,

• 𝑛 ≥ 2 and 𝜓 satisfies the conditions of Lemma 9,

there exist constants 𝐻𝑛 and ˆ𝐻𝑛 such that

and

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

(cid:18)∫ ∞

0

| 𝑓 ∗ Ψ𝜆 (𝑥)|2

𝑑𝜆
𝜆𝑛+1

(cid:19) 1/2(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)L1 (R𝑛)

≤ 𝑐 · 𝐻𝑛 ∥ 𝑓 ∥H1 (R𝑛)

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

(cid:32)

∑︁

𝑗 ∈Z

| 𝑓 ∗ Ψ𝑗 (𝑥)|2

(cid:33) 1/2(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)L1 (R𝑛)

≤ 𝑐 · ˆ𝐻𝑛 ∥ 𝑓 ∥H1 (R𝑛).

Now we can use the results above for our main dilation stability results.

Theorem 16. Suppose that 𝜓 is a wavelet that satisfies the conditions of Lemma 14. Then there

exists a constants 𝐾𝑛,𝑚 and ˆ𝐾𝑛,𝑚 only dependent on 𝑛 and 𝑚 such that

and

for any |𝑐| < 1
2𝑛 .

∥𝑆𝑚

cont,2

𝑓 − 𝑆𝑚

cont,2

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) ≤ |𝑐| · 𝐾𝑛,𝑚 ∥ 𝑓 ∥2

∥𝑆𝑚

dyad,2

𝑓 − 𝑆𝑚

dyad,2

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) ≤ |𝑐| · ˆ𝐾𝑛,𝑚 ∥ 𝑓 ∥2

33

Proof. Without loss of generality, assume 𝑐 > 0. Let

W𝑡 𝑓 = 𝑓 ∗ 𝜓𝑡

𝑀 𝑓 = | 𝑓 |

𝑈𝑡 = 𝑀W𝑡

𝐴𝑞 𝑓 =

(cid:18)∫

R𝑛

𝑓 𝑞 (𝑥) 𝑑𝑥

(cid:19) 1/𝑞

.

It follows that 𝑆𝑚

cont,2 = 𝐴2𝑀𝑊𝜆𝑚𝑈𝜆𝑚−1 · · · 𝑈𝜆1. We will also let 𝑉𝑚−1 = 𝑈𝜆𝑚−1 · · · 𝑈𝜆1, with 𝑉0
being the identity operator, and make a slight abuse of notation by denoting W𝜆𝑚 as W. First, we

will add and subtract 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 and apply triangle inequality:

∥𝑆𝑚

cont,2

𝑓 − 𝑆𝑚

cont,2

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) = ∥ 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀W𝑉𝑚−1𝐿𝜏 𝑓 ∥L2 (R𝑚
+ )

≤ ∥ 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥L2 (R𝑚
+ )

+ ).
+ ∥ 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 − 𝐴2𝑀W𝑉𝑚−1, 𝐿𝜏 𝑓 ∥L2 (R𝑚

We’ll start by bounding the first term. We see that 𝑔 = W𝑉𝑚−1 𝑓 ∈ L2(R𝑛). Thus

| 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 | = |∥𝑔∥2 − ∥𝐿𝜏𝑔∥2| .

Now use a change of variables:

∥𝐿𝜏𝑔∥2

2 =

∫

R𝑛

|𝑔((1 − 𝑐)𝑥)|2 𝑑𝑥 = (1 − 𝑐)−𝑛 ∥𝑔∥2
2

.

It then follows that

|∥𝐿𝜏𝑔∥2 − ∥𝑔∥2| = ∥𝑔∥2

(cid:18)

1
(1 − 𝑐)𝑛/2

(cid:19)

− 1

≤ ∥𝑔∥2

(cid:18)

1
(1 − 𝑐)𝑛 − 1

(cid:19)

.

34

Taking the scattering norm yields

∥ 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥2

L2 (R𝑚

+ ) ≤

=

(cid:19) 2

(cid:18)

1
(1 − 𝑐)𝑛 − 1
(cid:19) 2
(cid:18) 1 − (1 − 𝑐)𝑛
(1 − 𝑐)𝑛

∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R𝑚
+ )

∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R𝑚
+ )

1
(1 − 𝑐)𝑛

𝑛
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛
𝑗

𝑐 𝑗 (cid:170)
(cid:174)
(cid:172)

(cid:18) 2𝑛
2𝑛 − 1

(cid:19) 𝑛 𝑛
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛
𝑗

𝑐 𝑗

= (cid:169)
(cid:173)
(cid:171)







≤

2

∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R𝑚
+ )

2








∥𝑆𝑚

cont,2

𝑓 ∥2

L2 (R𝑚
+ )

≤ 𝑐2 · 𝐶𝑚,𝑛 ∥ 𝑓 ∥2
2

.

For the second term, apply Minkwoski’s inequality for 2 norms:

∥ 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 − 𝐴2𝑀W𝑉𝑚−1𝐿𝜏 𝑓 ∥L2 (R𝑚
+ )

(cid:32)∫ ∞

0
(cid:32)∫ ∞

∫ ∞

0

∫ ∞

· · ·

· · ·

0

0

=

≤

|∥𝐿𝜏W𝑉𝑚−1 𝑓 ∥2 − ∥W 𝐿𝜏𝑉𝑚−1 𝑓 ∥2|2

𝑑𝜆1
𝜆𝑛+1
1

∥𝐿𝜏W𝑉𝑚−1 𝑓 − W 𝐿𝜏𝑉𝑚−1 𝑓 ∥2
2

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

(cid:33) 1/2

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚
(cid:33) 1/2

= ∥ 𝐴2𝑀 [W𝑉𝑚−1, 𝐿𝜏] 𝑓 ∥L2 (R𝑚
+ ).

Now this is a commutator term, and we can now bound:

∥ 𝐴2𝑀 [W𝑉𝑚−1, 𝐿𝜏] 𝑓 ∥2

L2 (R𝑚

+ ) =

∫ ∞

∫ ∞

· · ·

0

0

∥ [W𝑉𝑚−1, 𝐿𝜏] 𝑓 ∥2
2

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

= ∥| [W𝑉𝑚−1, 𝐿𝜏] 𝑓 ∥2

≤ ∥ [W𝑉𝑚−1, 𝐿𝜏] ∥2

L2 (R𝑚
+ ×R𝑛)
+ ×R𝑛)→L2 (R𝑛) ∥ 𝑓 ∥2

2

L2 (R𝑚

.

We examine the commutator term more closely. Without a loss of generality, assume 𝑚 ≥ 2. By

expanding it, we see that each term contains [W, 𝐿𝜏]. It follows that

∥ [W𝑉𝑚−1, 𝐿𝜏] ∥L2 (R𝑚

≤ 𝑚∥W ∥𝑚−1

+ ×R𝑛)
L2 (R+×R𝑛)→L2 (R𝑛) ∥ 𝑀 ∥𝑚−1
≤ 𝐶𝑚 ∥ [W, 𝐿𝜏] ∥L2 (R+×R𝑛)→L2 (R𝑛).

L2 (R𝑛)→L2 (R𝑛) ∥ [W, 𝐿𝜏] ∥L2 (R+×R𝑛)→L2 (R𝑛)

35

Thus, once we bound this quantity appropriately, we will finish the proof. We start by writing

∥ [W, 𝐿𝜏] 𝑓 ∥2

L2 (R+×R𝑛) =

∫ ∞

0

∥(𝐿𝜏 𝑓 ) ∗ 𝜓𝜆 − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) ∥2
2

𝑑𝜆
𝜆𝑛+1

.

By substitution with 𝑧 = (1 − 𝑐)𝑥 and Lemma 13,

∥(𝐿𝜏 𝑓 ) ∗ 𝜓𝜆 − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) ∥2
2

∫

=

R𝑛

∫

R𝑛

(cid:12)
(cid:12)
=
(cid:12)
= (1 − 𝑐)−𝑛 ∫
= (1 − 𝑐)−𝑛 ∫
= (1 − 𝑐)−𝑛 ∫

R𝑛

R𝑛

R𝑛

|( 𝐿𝜏 𝑓 ∗ 𝜓𝜆 )(𝑥) − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) (𝑥)|2 𝑑𝑥

(1 − 𝑐)−𝑛/2 (cid:0) 𝑓 ∗ 𝜓(1−𝑐)𝜆(cid:1) ((1 − 𝑐)𝑥) − ( 𝑓 ∗ 𝜓𝜆) ((1 − 𝑐)𝑥)

(cid:12)
2
(cid:12)
(cid:12)

𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

(1 − 𝑐)−𝑛/2 (cid:0) 𝑓 ∗ 𝜓(1−𝑐)𝜆(cid:1) (𝑧) − ( 𝑓 ∗ 𝜓𝜆) (𝑧)

(cid:12)
2
(cid:12)
(cid:12)

𝑑𝑧

(cid:16)

𝑓 ∗

(1 − 𝑐)−𝑛/2𝜓(1−𝑐)𝜆 − 𝜓𝜆

(cid:17)(cid:12)
2
(cid:12)
(cid:12)

𝑑𝑧

|( 𝑓 ∗ Ψ𝜆) (𝑧)|2 𝑑𝑧,

= (1 − 𝑐)−𝑛 ∥ 𝑓 ∗ Ψ𝜆 ∥2
2

.

Thus, we obtain

∫ ∞

0

∥(𝐿𝜏 𝑓 ) ∗ 𝜓𝜆 − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) ∥2
2

𝑑𝜆
𝜆𝑛+1

∥ 𝑓 ∗ Ψ𝜆 ∥2
2

𝑑𝜆
𝜆𝑛+1

∫ ∞

| 𝑓 ∗ Ψ𝜆 (𝑥)|2

𝑑𝜆
𝜆𝑛+1

𝑑𝑥

| 𝑓 ∗ Ψ𝜆 (𝑥)|2

𝑑𝜆
𝜆𝑛+1

(cid:19) 1/2(cid:13)
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)
2

0

= (1 − 𝑐)−𝑛 ∫ ∞
= (1 − 𝑐)−𝑛 ∫
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:18) 2𝑛
2𝑛 − 1

= (1 − 𝑐)−𝑛

≤ 𝑐2 ·

0
(cid:19) 𝑛

R𝑛
0
(cid:18)∫ ∞

𝐶𝑛,𝑝 ∥ 𝑓 ∥2
2

.

It follows that

for any 𝑐 < 1
2𝑛 .

∥𝑆𝑚

cont,2

𝑓 − 𝑆𝑚

cont,2

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) ≤ |𝑐| · 𝐾𝑛,𝑚 ∥ 𝑓 ∥2

□

As is customary at this point, we have the following corollaries. We start with the case where

1 < 𝑞 < 2.

36

Corollary 17. Assume |𝑐| < 1

2𝑛 . For 𝑞 ∈ (1, 2), there exists constants 𝐾𝑛,𝑚,𝑞 and ˆ𝐾𝑛,𝑚,𝑞 such that

∥𝑆𝑚

cont,𝑞 𝑓 − 𝑆𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

𝑞
L2 (R𝑚
+ )

≤ |𝑐|𝑞 · 𝐾𝑛,𝑚,𝑞 ∥ 𝑓 ∥

𝑞
𝑞

and

∥𝑆𝑚

dyad,𝑞 𝑓 − 𝑆𝑚

dyad,𝑞 𝐿𝜏 𝑓 ∥

𝑞
ℓ2 (Z𝑚)

≤ |𝑐|𝑞 · ˆ𝐾𝑛,𝑚,𝑞 ∥ 𝑓 ∥

𝑞
𝑞.

Proof. Without loss of generality again, assume 𝑐 > 0. First, we will add and subtract 𝐴𝑞 𝑀 𝐿𝜏W𝑉𝑚−1 𝑓

and apply triangle inequality:

∥𝑆𝑚

cont,𝑞 𝑓 − 𝑆𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) = ∥ 𝐴𝑞 𝑀W𝑉𝑚−1 𝑓 − 𝐴𝑞 𝑀W𝑉𝑚−1𝐿𝜏 𝑓 ∥L2 (R𝑚
+ )

≤ ∥ 𝐴𝑞 𝑀W𝑉𝑚−1 𝑓 − 𝐴𝑞 𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥L2 (R𝑚
+ )

+ ∥ 𝐴𝑞 𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 − 𝐴𝑞 𝑀W𝑉𝑚−1, 𝐿𝜏 𝑓 ∥L2 (R𝑚
+ ).

We’ll start by bounding the first term again. Define 𝑔 = W𝑉𝑚−1 𝑓 ∈ L𝑞 (R𝑛). and we have

| 𝐴𝑞 𝑀W𝑉𝑚−1 𝑓 − 𝐴𝑞 𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 | = (cid:12)

(cid:12)∥𝑔∥𝑞 − ∥𝐿𝜏𝑔∥𝑞

(cid:12)
(cid:12) .

By change of variables,

(cid:12)
(cid:12)∥𝑔∥𝑞 − ∥𝐿𝜏𝑔∥𝑞

(cid:12)
(cid:12) = ∥𝑔∥𝑞

(cid:18)

(cid:19)

1
(1 − 𝑐)𝑛/𝑞 − 1

≤ ∥𝑔∥𝑞

(cid:18)

1
(1 − 𝑐)𝑛 − 1

(cid:19)

.

Again, we have

∥ 𝐴𝑞 𝑀W𝑉𝑚−1 𝑓 − 𝐴𝑞 𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥

𝑞
L2 (R𝑚
+ )

≤

≤

=

=

≤

(cid:19) 𝑞

∥𝑆𝑚

cont,2

𝑓 ∥

𝑞
L2 (R𝑚
+ )

∥𝑆𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚
+ )

∥𝑆𝑚

cont,𝑞 𝑓 ∥
𝑞

𝑞
L2 (R𝑚
+ )

(cid:18)

(cid:18)

1
(1 − 𝑐)𝑛/𝑞 − 1
(cid:19) 𝑞

1
(1 − 𝑐)𝑛 − 1
(cid:21) 𝑞
(cid:20) 1 − (1 − 𝑐)𝑛
(1 − 𝑐)𝑛
𝑛
∑︁

1
(1 − 𝑐)𝑛

𝑗=1

(cid:19)

(cid:18)𝑛
𝑗

𝑐 𝑗








(cid:18) 2𝑛
2𝑛 − 1

(cid:19) 𝑛 𝑛
∑︁

𝑗=1

(cid:19)

(cid:18)𝑛
𝑗

𝑐 𝑗

≤ |𝑐|𝑞 · 𝐶𝑚,𝑛 ∥ 𝑓 ∥

𝑞
𝑞.

∥𝑆𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚
+ )

𝑞








∥𝑆𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚
+ )














37

For the second term, apply Minkoski’s inequality for 𝑞 norms:

∥ 𝐴𝑞 𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 − 𝐴𝑞 𝑀W𝑉𝑚−1, 𝐿𝜏 𝑓 ∥L2 (R𝑚
+ )

(cid:32)∫ ∞

0
(cid:32)∫ ∞

∫ ∞

0

∫ ∞

· · ·

· · ·

0

0

=

≤

(cid:12)
(cid:12)∥𝐿𝜏W𝑉𝑚−1 𝑓 ∥𝑞 − ∥W 𝐿𝜏𝑉𝑚−1 𝑓 ∥𝑞

2 𝑑𝜆1
(cid:12)
(cid:12)
𝜆𝑛+1
1

∥𝐿𝜏W𝑉𝑚−1 𝑓 − W 𝐿𝜏𝑉𝑚−1 𝑓 ∥2
𝑞

𝑑𝜆1
𝜆𝑛+1
1

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

(cid:33) 1/2

· · ·

𝑑𝜆𝑚
𝜆𝑛+1
𝑚
(cid:33) 1/2

+ ).
= ∥ 𝐴𝑞 𝑀 [W𝑉𝑚−1, 𝐿𝜏] 𝑓 ∥L2 (R𝑚

Via a similar reduction technique for Theorem 16, we can reduce to a commutator bound above.

Additionally, we have

Thus,

∥(𝐿𝜏 𝑓 ) ∗ 𝜓𝜆 − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) ∥

𝑞
𝑞 = (1 − 𝑐)−𝑛 ∥ 𝑓 ∗ Ψ𝜆 ∥

𝑞
𝑞 .

∥ 𝐴𝑞 𝑀 [W, 𝐿𝜏] 𝑓 ∥

𝑞
L2 (R𝑚
+ )

=

(cid:18)∫ ∞

0

∥(𝐿𝜏 𝑓 ) ∗ 𝜓𝜆 − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) ∥2
𝑞

(cid:19) 𝑞/2

𝑑𝜆
𝜆𝑛+1

(cid:18)∫ ∞

= (1 − 𝑐)−𝑛

∥ 𝑓 ∗ Ψ𝜆 ∥2
𝑞

(cid:19) 𝑞/2

𝑑𝜆
𝜆𝑛+1

≤ (1 − 𝑐)−𝑛

0
(cid:18)∫ ∞

0

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤ |𝑐|𝑞 · ˜𝐶𝑛 ∥ 𝑓 ∥

𝑞
𝑞.

| 𝑓 ∗ Ψ𝜆 (𝑥)|2

𝑑𝜆
𝜆𝑛+1

(cid:19) 1/2(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑞

𝑞

It follows that

for any |𝑐| < 1
2𝑛 .

∥𝑆𝑚

cont,𝑞 𝑓 − 𝑆𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

𝑞
L2 (R𝑚
+ )

≤ |𝑐|𝑞 · 𝐾𝑛,𝑚 ∥ 𝑓 ∥

𝑞
𝑞

□

Additionally, for the case of 𝑞 = 1, we have the following corollary that we will state, but not

prove, since the idea is the same as the previous corollary.

Corollary 18. Suppose one of the following holds:

• 𝑛 = 1, 𝜓 is complex analytic and satisfies the conditions of Lemma 14,

38

• 𝑛 ≥ 2 and 𝜓 satisfies the conditions of Lemma 9,

then there exist constants 𝐾𝐻,𝑚 and ˆ𝐾𝐻,𝑚 such that

∥𝑆𝑚

cont,1

𝑓 − 𝑆𝑚

cont,1

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) ≤ 𝑐 · 𝐾𝐻,𝑚 ∥ 𝑓 ∥H1 (R𝑛)

and

∥𝑆𝑚

dyad,1

𝑓 − 𝑆𝑚

dyad,1

𝐿𝜏 𝑓 ∥ℓ2 (Z𝑚) ≤ 𝑐 · ˆ𝐾𝐻,𝑚 ∥ 𝑓 ∥H1 (R𝑛).

2.4 Stability to Diffeomorphisms

We now focus on the stability of 𝑆𝑚

cont,𝑞 𝑓 for general diffeomorphisms with ∥𝐷𝜏∥∞ < 1

2𝑛 . The

corresponding operator for diffeomorphisms is defined as 𝐿𝜏 𝑓 (𝑥) = 𝑓 (𝑥 − 𝜏(𝑥)).

2.4.1 Stability to Diffeomorphisms When 𝑞 = 2

Proposition 19. Assume 𝜓 and its first and second order derivatives have decay∗ in 𝑂 ((1+|𝑥|)−𝑛−3),
and ∫

R𝑛 𝜓(𝑥) 𝑑𝑥 = 0. Then for every 𝜏 ∈ 𝐶2(R𝑛) with ∥𝐷𝜏∥∞ ≤ 1

2𝑛 , there exists ˜𝐶𝑛 > 0 such that:

∥ [W, 𝐿𝜏] ∥L2 (R+×R𝑛)→L2 (R𝑛) ≤ ˜𝐶𝑛

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19)

.

Proof. The argument is a continuous version of Lemma 2.14 in [11]. We will first show how to

transform our commutator term into an analogous commutator term from [11]. To shorten notation,

we will denote ∥ [W, 𝐿𝜏] ∥L2 (R+×R𝑛) as ∥ [W, 𝐿𝜏] ∥. We have

∥ [W, 𝐿𝜏] 𝑓 ∥2

L2 (R+×R𝑛) =

=

=

∫ ∞

0
∫ ∞

0
∫ ∞

∫

0

R𝑛

∥ [W𝑡, 𝐿𝜏] 𝑓 ∥2
2

𝑑𝑡
𝑡𝑛+1

∥𝜓𝑡 ∗ (𝐿𝜏 𝑓 ) − 𝐿𝜏 (𝜓𝑡 ∗ 𝑓 ) ∥2
2

𝑑𝑡
𝑡𝑛+1

|𝜓𝑡 ∗ (𝐿𝜏 𝑓 ) − 𝐿𝜏 (𝜓𝑡 ∗ 𝑓 )|2 𝑑𝑥

𝑑𝑡
𝑡𝑛+1

.

Notice that 𝜓 1

𝑡

(𝑥) = 𝑡𝑛/2𝜓(𝑡𝑥). Use the change of variables 𝑡 = 1

𝜆 to get

∥ [W, 𝐿𝜏] 𝑓 ∥2

L2 (R+×R𝑛) =

=

∫ ∞

0
∫ ∞

0

𝜆

𝜓 1

(cid:13)
(cid:13)
(cid:13)
(cid:13)
𝜆𝑛/2𝜓 1
(cid:13)
(cid:13)

𝜆

∗ (𝐿𝜏 𝑓 ) − 𝐿𝜏 (𝜓 1

∗ 𝑓 )

(cid:13)
(cid:13)
(cid:13)

2

2

𝜆𝑛−1 𝑑𝜆

𝜆

∗ (𝐿𝜏 𝑓 ) − 𝐿𝜏 (𝜆𝑛/2𝜓 1

𝜆

∗ 𝑓 )

(cid:13)
2
(cid:13)
(cid:13)
2

𝑑𝜆
𝜆

.

∗Similar to [31], we have found that there needs to be 𝑂 ((1 + |𝑥|) −𝑛−2+𝛼) decay for some 𝛼 > 0 to bound (E.26)

in [11].

39

Define 𝒲𝜆 𝑓 = 𝑓 ∗ 𝜆𝑛/2𝜓 1

𝜆

with 𝜆𝑛/2𝜓 1

𝜆

(𝑥) = 𝜆𝑛𝜓(𝜆𝑥). In other words, 𝒲𝑡 is a convolution with an

L1 normalized wavelet, which matches with the normalization in [11]. Now we have

∥ [W, 𝐿𝜏] 𝑓 ∥2

L2 (R+×R𝑛) =

∫ ∞

0

∥ [𝒲𝜆, 𝐿𝜏] 𝑓 ∥2
2

𝑑𝜆
𝜆

.

This implies

[W, 𝐿𝜏]∗ [W, 𝐿𝜏] =

∫ ∞

[𝒲𝜆, 𝐿𝜏]∗ [𝒲𝜆, 𝐿𝜏]

Defining 𝐾𝜆 = 𝒲𝜆 − 𝐿𝜏𝒲𝜆 𝐿−1

0
𝜏 so that [𝒲𝜆, 𝐿𝜏] = 𝐾𝜆 𝐿𝜏, we have:

𝑑𝜆
𝜆

∥ [W, 𝐿𝜏] ∥ = ∥ [W, 𝐿𝜏]∗ [W, 𝐿𝜏] ∥1/2

=

=

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)
0
(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

0

[𝒲𝜆, 𝐿𝜏]∗ [𝒲𝜆, 𝐿𝜏]

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

𝐿∗
𝜏𝐾 ∗

𝜆𝐾𝜆 𝐿𝜏

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)
𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

1/2

,

≤ ∥𝐿𝜏 ∥ ·

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

0

𝐾 ∗

𝜆𝐾𝜆

Since ∥𝐿𝜏 𝑓 ∥2

2 ≤

(cid:16)

1
1−𝑛∥𝐷𝜏∥∞

(cid:17)

∥ 𝑓 ∥2
2,

∥𝐿𝜏 ∥ ≤

1
1 − 𝑛∥𝐷𝜏∥∞

≤ 2

and the problem is reduced to bounding

(cid:13)
(cid:13)
(cid:13)

∫ ∞
0

𝐾 ∗

𝜆𝐾𝜆 𝜆−1 𝑑𝜆

1/2

(cid:13)
(cid:13)
(cid:13)

. Let 𝛾 ≥ 1. The integral is divided

into three pieces:

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

0

𝐾 ∗

𝜆𝐾𝜆

1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤

≤

∫ 2−𝛾

(cid:18)(cid:13)
(cid:13)
(cid:13)
(cid:13)
0
∫ 2−𝛾

(cid:13)
(cid:13)
(cid:13)
(cid:13)

0

𝐾 ∗

𝜆𝐾𝜆

𝐾 ∗

𝜆𝐾𝜆

+

(cid:13)
𝑑𝜆
(cid:13)
(cid:13)
𝜆
(cid:13)
(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

+

∫ 1

2−𝛾
(cid:13)
∫ 1
(cid:13)
(cid:13)
(cid:13)

2−𝛾

𝐾 ∗

𝜆𝐾𝜆

𝐾 ∗

𝜆𝐾𝜆

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)
1/2

1

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)
𝑑𝜆
𝜆

+

(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝐾 ∗

𝜆𝐾𝜆

(cid:19) 1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

+

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

1

𝐾 ∗

𝜆𝐾𝜆

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

= 𝑃1 + 𝑃2 + 𝑃3.

To bound 𝑃1, we decompose 𝐾𝜆 = ˜𝐾𝜆,1 + ˜𝐾𝜆,2, where the kernels defining ˜𝐾𝜆,1, ˜𝐾𝜆,2 are

˜𝑘𝜆,1(𝑥, 𝑢) := (1 − det(𝐼 − 𝐷𝜏(𝑢)))𝜆𝑛𝜓(𝜆(𝑥 − 𝑢))

:= 𝑎(𝑢)𝜆𝑛𝜓(𝜆(𝑥 − 𝑢)),

˜𝑘𝜆,2(𝑥, 𝑢) := det(𝐼 − 𝐷𝜏(𝑢))(𝜆𝑛𝜓(𝜆(𝑥 − 𝑢)) − 𝜆𝑛𝜓(𝜆(𝑥 − 𝜏(𝑥) − 𝑢 + 𝜏(𝑢))),

40

respectively. Since our normalization matches with [11], E.13 implies that there exists a constant

𝐶𝑛 such that

We want to prove that

∥ ˜𝐾𝜆,2∥ ≤ 𝐶𝑛𝜆∥Δ𝜏∥∞.

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∫ 1

0

˜𝐾 ∗

𝜆,1

˜𝐾𝜆,1

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

≤ 𝐶𝑛 ∥𝐷𝜏∥∞.

Let 𝑓 ∈ L2(R𝑛) be arbitrary and define ˜𝜓(𝑡) = 𝜓∗(−𝑡). Based on [11], the kernel of ˜𝐾 ∗
𝜆,1

˜𝐾𝜆,1 is

given by

˜𝑘𝜆 (𝑦, 𝑧) := 𝑎(𝑦)𝑎(𝑧)𝜆𝑛/2 ˜𝜓 1
𝜆

∗ 𝜆𝑛/2 ˜𝜓 1
𝜆

(𝑧 − 𝑦).

Thus, it is sufficient to bound the quantity

∫ 1

0

∥ ˜𝐾 ∗
𝜆,1

˜𝐾𝜆,1 𝑓 ∥2
2

𝑑𝜆
𝜆

.

We see that ∥𝑎∥∞ ≤ 𝑛∥𝐷𝜏∥∞. Substituting in the kernel and bounding yields

∫ 1

0

∥ ˜𝐾 ∗
𝜆,1

˜𝐾𝜆,1 𝑓 ∥2
2

𝑑𝜆
𝜆

=

=

∫ 1

∫

0
∫ 1

R𝑛

∫

0

R𝑛

(cid:18)

𝑎(𝑦)𝑎(𝑧)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∫

R𝑛

𝜆𝑛/2 ˜𝜓 1
𝜆

∗ 𝜆𝑛/2𝜓 1
𝜆

(cid:19)

(𝑧 − 𝑦) 𝑓 (𝑦) 𝑑𝑦

2

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑧

|𝑎(𝑧)|2

𝑎(𝑦)

∫

(cid:12)
(cid:12)
(cid:12)
(cid:12)
∫

R𝑛
(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)

R𝑛

R𝑛

(cid:18)

𝜆𝑛/2 ˜𝜓 1
𝜆

∗ 𝜆𝑛/2𝜓 1
𝜆

(cid:19)

(𝑧 − 𝑦) 𝑓 (𝑦) 𝑑𝑦

2

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:18)

𝑎(𝑦)

(cid:19)

𝜆𝑛/2 ˜𝜓 1
𝜆

∗ 𝜆𝑛/2𝜓 1
𝜆

(𝑧 − 𝑦) 𝑓 (𝑦) 𝑑𝑦

(cid:12)
2
(cid:12)
(cid:12)
(cid:12)

𝑑𝜆
𝜆

𝑑𝑧

𝑑𝜆
𝜆

.

≤ 𝑛2∥𝐷𝜏∥2
∞

∫ 1

0

𝑑𝜆
𝜆

𝑑𝑧

Let 𝐹 (𝑦) = 𝑎(𝑦) 𝑓 (𝑦) ∈ L2(R𝑛) and let F represent taking the Fourier Transform. Then we

substitute 𝐹 (𝑦) for 𝑎(𝑦) 𝑓 (𝑦) in the last line of the inequality above to get

𝑛2∥𝐷𝜏∥2
∞

∫ 1

∫

(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)
R𝑛
∫
∫ 1

0

= 𝑛2∥𝐷𝜏∥2
∞

= 𝑛2∥𝐷𝜏∥2
∞

= 𝑛2∥𝐷𝜏∥2
∞

0
∫ 1

0
∫

R𝑛

R𝑛
(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

F

R𝑛
(cid:18)

R𝑛

∫

R𝑛

𝑎(𝑦)

(cid:18)

𝜆𝑛/2 ˜𝜓 1
𝜆

(cid:18)

𝜆𝑛/2 ˜𝜓 1
𝜆

∗ 𝜆𝑛/2𝜓 1
𝜆
(cid:19)

∗ 𝜆𝑛/2𝜓 1
𝜆
(cid:19)

𝜆𝑛/2 ˜𝜓 1
𝜆

∗ 𝜆𝑛/2𝜓 1
𝜆
(cid:19)
𝑑𝜆
𝜆

𝜆 )|4

| ˆ𝜓( 𝜔

| ˆ𝐹 (𝜔)|2

(cid:18)∫ 1

0

(cid:12)
2
(cid:12)
(cid:12)
(cid:12)

𝑑𝑧

𝑑𝜆
𝜆

𝑑𝑧

𝑑𝜆
𝜆

(cid:19)

(𝑧 − 𝑦) 𝑓 (𝑦) 𝑑𝑦

2

(cid:12)
(cid:12)
(cid:12)
(cid:12)
𝑑𝜆
𝜆

(𝑧 − 𝑦)𝐹 (𝑦) 𝑑𝑦

(𝜔) ˆ𝐹 (𝜔)

2

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑧

𝑑𝜔.

To finish up the argument, we make a substitution to rewrite

∫ 1

0

| ˆ𝜓( 𝜔

𝜆 )|4

𝑑𝜆
𝜆

=

∫ ∞

1

| ˆ𝜓(𝜆𝜔)|4

𝑑𝜆
𝜆

.

41

Using our decay assumptions on 𝜓 and its partial derivatives, from Problem 6.1.3 in [38], we know

that

| ˆ𝜓(𝜔)| ≤ 𝑀𝜓min{|𝜔|, |𝜔|−2}

for some constant 𝑀𝜓. Now, consider the quantity ∫ ∞

0

| ˆ𝜓(𝜆𝜔)|4 𝑑𝜆

𝜆 . Without loss of generality,

assume that |𝜔| = 1 since dilations of 𝜔 do not change the integral. It follows that

∫ ∞

0

| ˆ𝜓(𝜆𝜔)|4

𝑑𝜆
𝜆

≤ 𝑀𝜓

∫ 1

0

𝜆3𝑑𝜆 + 𝑀𝜓

∫ ∞

1

𝜆−9𝑑𝜆 < ∞,

and we conclude that

∫ ∞

1

| ˆ𝜓(𝜆𝜔)|4

𝑑𝜆
𝜆

≤ 𝐴𝜓

for some constant 𝐴𝜓. To finish up,

𝑛2∥𝐷𝜏∥2
∞

∫

R𝑛

| ˆ𝐹 (𝜔)|2

(cid:18)∫ 1

0

| ˆ𝜓( 𝜔

𝜆 )|4

(cid:19)

𝑑𝜆
𝜆

𝑑𝜔 ≤ 𝑛2∥𝐷𝜏∥2

∞ 𝐴𝜓

≤ 𝑛2∥𝐷𝜏∥2

∞ 𝐴𝜓

∫

R𝑛

∫

R𝑛

| ˆ𝐹 (𝜔)|2 𝑑𝜔

|𝑎(𝑧) 𝑓 (𝑧)|2 𝑑𝑧

≤ 𝑛2∥𝐷𝜏∥2

∞ 𝐴𝜓 ∥ 𝑓 ∥2
2

.

Thus, we have the desired bound on

(cid:13)
∫ 1
(cid:13)
(cid:13)
0
(cid:13)

˜𝐾 ∗

𝜆,1

˜𝐾𝜆,1

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

.

42

Substituting everything in yields

∫ 2−𝛾
(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝐾 ∗

𝜆𝐾𝜆

1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

0
∫ 2−𝛾
(cid:13)
(cid:13)
(cid:13)
(cid:13)
0
∫ 2−𝛾
(cid:13)
(cid:13)
(cid:13)
(cid:13)
0
∫ 2−𝛾
(cid:18)(cid:13)
(cid:13)
(cid:13)
(cid:13)
0
∫ 2−𝛾
(cid:18)(cid:13)
(cid:13)
(cid:13)
(cid:13)
0
∫ 2−𝛾
(cid:13)
(cid:13)
(cid:13)
(cid:13)

0

=

=

≤

≤

≤

(cid:32)

(cid:16)

≤ 2𝐶𝑛

≤ 2𝐶𝑛

( ˜𝐾𝜆,1 + ˜𝐾𝜆,2)∗( ˜𝐾𝜆,1 + ˜𝐾𝜆,2)

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

( ˜𝐾 ∗
𝜆,1

˜𝐾𝜆,1 + ˜𝐾 ∗
𝜆,1

˜𝐾𝜆,2 + ˜𝐾 ∗
𝜆,2

˜𝐾𝜆,1 + ˜𝐾 ∗
𝜆,2

˜𝐾𝜆,2)

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

+

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
1/2

+

˜𝐾 ∗

𝜆,1

˜𝐾𝜆,1

˜𝐾 ∗

𝜆,1

˜𝐾𝜆,1

𝑑𝜆
𝜆

𝑑𝜆
𝜆

˜𝐾 ∗

𝜆,1

˜𝐾𝜆,1

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∥𝐷𝜏∥∞ + ∥Δ𝜏∥∞

∫ 2−𝛾
(cid:13)
(cid:13)
(cid:13)
(cid:13)
0
∫ 2−𝛾

0
(cid:18)∫ 2−𝛾

+

0
(cid:18)∫ 2−𝛾

0

(cid:19) 1/2

(cid:13)
(cid:13)
(cid:13)
(cid:13)
𝑑𝜆
𝜆

(cid:19) 1/2

˜𝐾 ∗

𝜆,1

˜𝐾𝜆,2 + ˜𝐾 ∗
𝜆,2

˜𝐾𝜆,1 + ˜𝐾 ∗
𝜆,2

˜𝐾𝜆,2

𝑑𝜆
𝜆

∥ ˜𝐾𝜆,2∥2

𝑑𝜆
𝜆

+

∥ ˜𝐾𝜆,2∥2

𝑑𝜆
𝜆

∫ 2−𝛾

0
(cid:19) 1/2

+

2∥ ˜𝐾𝜆,1∥∥ ˜𝐾𝜆,2∥

(cid:18)∫ 2−𝛾

0

2∥ ˜𝐾𝜆,1∥∥ ˜𝐾𝜆,2∥

(cid:19) 1/2

𝜆2

𝑑𝜆
𝜆

+ ∥𝐷𝜏∥1/2

∞ ∥Δ𝜏∥1/2
∞

(cid:18)∫ 2−𝛾

2𝜆

0

𝑑𝜆
𝜆

(cid:19) 1/2

𝑑𝜆
𝜆
(cid:19) 1/2(cid:33)

∥𝐷𝜏∥∞ + 2−𝛾 ∥Δ𝜏∥∞ + 2−𝛾/2∥𝐷𝜏∥1/2

∞ ∥Δ𝜏∥1/2
∞

(cid:17)

≤ 4𝐶𝑛 (∥𝐷𝜏∥∞ + 2−𝛾 ∥Δ𝜏∥∞) .

To bound 𝑃3, we decompose 𝐾𝜆 = 𝐾𝜆,1 + 𝐾𝜆,2, where the kernels defining 𝐾𝜆,1, 𝐾𝜆,2 are

𝑘𝜆,1(𝑥, 𝑢) = 𝜆𝑛𝜓(𝜆(𝑥 − 𝑢)) − 𝜆𝑛𝜓(𝜆(𝐼 − 𝐷𝜏(𝑢)) (𝑥 − 𝑢)) det(𝐼 − 𝐷𝜏(𝑢))

𝑘𝜆,2(𝑥, 𝑢) = det(𝐼 − 𝐷𝜏(𝑢))𝜆𝑛𝜓(𝜆(𝐼 − 𝐷𝜏(𝑢)) (𝑥 − 𝑢)) − 𝜆𝑛𝜓(𝜆(𝑥 − 𝜏(𝑥) − 𝑢 + 𝜏(𝑢))).

A similar computation to the one for 𝑃1 shows that:

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

1

𝐾 ∗

𝜆𝐾𝜆

1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤

+

∫ ∞

(cid:13)
(cid:13)
(cid:13)
(cid:13)
1
(cid:18)∫ ∞

𝐾 ∗

𝜆,1

𝐾𝜆,1

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

+

2∥𝐾𝜆,1∥∥𝐾𝜆,2∥

𝑑𝜆
𝜆

1

∥𝐾𝜆,2∥2

(cid:19) 1/2

𝑑𝜆
𝜆

(cid:18)∫ ∞

1
(cid:19) 1/2

.

Letting 𝑄 𝑗 = 𝐾 ∗

2 𝑗 ,1

𝐾2 𝑗 ,1, it is shown in [11] that:

∥𝐾𝜆,1∥ ≤ 𝐶𝑛 ∥𝐷𝜏∥∞

∥𝐾𝜆,2∥ ≤ min{𝜆−𝑛 ∥𝐷2𝜏∥∞, ∥𝐷𝜏∥∞}

∥𝑄 𝑗 𝑄ℓ ∥ ≤ 𝐶2

𝑛2−| 𝑗−ℓ| (∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞)4

43

so that

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

1

𝐾 ∗

𝜆,1

𝐾𝜆,1

1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

=

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)
= √︁

0

log(2)

𝐾 ∗

2 𝑗 ,1

𝐾2 𝑗 ,1 log(2) 𝑑𝑗

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∫ ∞

0

𝑄 𝑗 𝑑𝑗

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

.

We now apply a continuous version of Cotlar’s Lemma (see Ch. 7 of [42], Sec. 5.5 for the

continuous extension). We define:

𝛽( 𝑗, ℓ) =

𝐶𝑛2−| 𝑗−ℓ|/2(∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞)2

𝑗 ≥ 0 and ℓ ≥ 0

.

0

otherwise





Defining 𝑄 𝑗 = 0 for 𝑗 < 0, we have ∥𝑄∗

𝑗 𝑄ℓ ∥ ≤ 𝛽( 𝑗, ℓ)2 and ∥𝑄 𝑗 𝑄∗

ℓ ∥ ≤ 𝛽( 𝑗, ℓ)2 for all 𝑗, ℓ. Thus

∫

by Cotlar’s Lemma:
(cid:13)
(cid:13)
(cid:13)
(cid:13)
R
(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

0

𝑄 𝑗 𝑑 𝑗

𝑄 𝑗 𝑑 𝑗

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤ sup
𝑗 ∈R

≤ sup
𝑗 ≥0

∫

R
∫ ∞

0

𝛽( 𝑗, ℓ) 𝑑ℓ,

𝛽( 𝑗, ℓ) 𝑑ℓ

≤ 𝐶𝑛 (∥𝐷𝜏∥∞ + ∥𝐻𝜏∥∞)2

(cid:32)

sup
𝑗 ≥0

∫ ∞

0

2−| 𝑗−ℓ|/2 𝑑ℓ

(cid:33)

.

Now observing that with the change of variable 𝜆1 = 2 𝑗 , 𝜆2 = 2ℓ, we have 2−| 𝑗−ℓ|/2 = 𝜆1
𝜆2

∧ 𝜆2
𝜆1

, we

obtain:

∫ ∞

0

sup
𝑗 ≥0

2−| 𝑗−ℓ|/2 𝑑ℓ = sup
𝜆1≥1

∫ ∞

1

𝑑𝜆2
ln(2)𝜆2

(𝜆1 ∧ 𝜆2)
√
𝜆1𝜆2
(cid:32)∫ 𝜆1

√

(2

∫ ∞

𝑑𝜆2 +

1
𝜆1𝜆2
𝜆1
√︁𝜆1 − 2) + √︁𝜆1
(cid:19)

(cid:33)

𝑑𝜆2

(cid:19)(cid:19)

√

𝜆1
𝜆3/2
2
(cid:18) 2
√
𝜆1

sup
𝜆1≥1

sup
𝜆1≥1

sup
𝜆1≥1

1
(cid:18) 1
√
𝜆1

(cid:18)

4 −

2
√
𝜆1

=

=

=

=

1
ln(2)

1
ln(2)
1
ln(2)
4
ln(2)

and conclude that

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∫ ∞

1

𝐾 ∗

𝜆,1

𝐾𝜆,1

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

≤ 3𝐶𝑛 (∥𝐷𝜏∥∞ + ∥𝐻𝜏∥∞).

44

Thus we have:

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

1

𝐾 ∗

𝜆𝐾𝜆

1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤

+

∫ ∞

(cid:13)
(cid:13)
(cid:13)
(cid:13)
1
(cid:18)∫ ∞

𝐾 ∗

𝜆,1

𝐾𝜆,1

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

+

2∥𝐾𝜆,1∥∥𝐾𝜆,2∥

𝑑𝜆
𝜆

1

∥𝐾𝜆,2∥2

(cid:19) 1/2

𝑑𝜆
𝜆

(cid:18)∫ ∞

1
(cid:19) 1/2

.

Now we see that there exists a constant 𝐶𝑛 such that

∫ ∞

(cid:13)
(cid:13)
(cid:13)
(cid:13)
1
(cid:18)∫ ∞

1

𝐾 ∗

𝜆,1

𝐾𝜆,1

∥𝐾𝜆,2∥2

2∥𝐾𝜆,1∥∥𝐾𝜆,2∥

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)
(cid:19) 1/2

(cid:19) 1/2

𝑑𝜆
𝜆

𝑑𝜆
𝜆

𝑑𝜆
𝜆

(cid:18)∫ ∞

1

and

≤ 𝐶𝑛 (∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞)

≤ 𝐶𝑛 ∥𝐷2𝜏∥∞

(cid:18)∫ ∞

1

(cid:19) 1/2

𝜆−2𝑛 𝑑𝜆
𝜆
(cid:18)∫ ∞

≤ 𝐶𝑛 ∥𝐷𝜏∥1/2

∞ ∥𝐷2𝜏∥1/2
∞

2𝜆−𝑛 𝑑𝜆
𝜆

(cid:19) 1/2

.

1

(cid:13)
∫ ∞
(cid:13)
(cid:13)
(cid:13)

1

𝐾 ∗

𝜆𝐾𝜆

1/2

𝑑𝜆
𝜆

(cid:13)
(cid:13)
(cid:13)
(cid:13)

(cid:18)

(cid:18)

≤ 𝐶𝑛

≤ 𝐶𝑛

∥𝐷𝜏∥∞ +

∥𝐷𝜏∥∞ +

1
2𝑛
1
2𝑛

∥𝐷2𝜏∥∞ +

∥𝐷2𝜏∥∞ +

2
𝑛

1
𝑛

∥𝐷𝜏∥1/2

∞ ∥𝐷2𝜏∥1/2
∞

(cid:19)

∥𝐷𝜏∥∞ +

(cid:19)

∥𝐷2𝜏∥∞

1
𝑛

≤ 2𝐶𝑛 (∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞).

Finally, we bound 𝑃2. Note that in the previous section it was observed (shown in [11]) that

∥𝐾𝜆,1∥ ≤ 𝐶𝑛 ∥𝐷𝜏∥∞

∥𝐾𝜆,2∥ ≤ min{𝜆−𝑛 ∥𝐷2𝜏∥∞, ∥𝐷𝜏∥∞}.

The above two inequalities imply

∥𝐾𝜆 ∥ = ∥𝐾𝜆,1 + 𝐾𝜆,2∥ ≤ ∥𝐾𝜆,1∥ + ∥𝐾𝜆,2∥ ≤ 2𝐶𝑛 ∥𝐷𝜏∥∞

so that

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∫ 1

2−𝛾

𝐾 ∗

𝜆𝐾𝜆

𝑑𝜆
𝜆

(cid:13)
1/2
(cid:13)
(cid:13)
(cid:13)

≤

(cid:18)∫ 1

2−𝛾

∥𝐾𝜆 ∥2

(cid:19) 1/2

𝑑𝜆
𝜆

≤ 2𝐶𝑛 ∥𝐷𝜏∥∞

(cid:19) 1/2

(cid:18)∫ 1

2−𝛾

𝑑𝜆
𝜆

≤ 2𝐶𝑛 ∥𝐷𝜏∥∞ (− ln(2−𝛾))1/2

≤ 2𝐶𝑛𝛾1/2∥𝐷𝜏∥∞.

45

Putting everything together and since 𝛾 ≥ 1, we obtain:

∥ [W, 𝐿𝜏] ∥ ≤ 2(𝑃1 + 𝑃2 + 𝑃3)

≤ 4𝐶𝑛 (∥𝐷𝜏∥∞ + 2−𝛾 ∥Δ𝜏∥∞) + 2𝐶𝑛𝛾1/2∥𝐷𝜏∥∞ + 3𝐶𝑛 (∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞)

(cid:16)

≤ ˜𝐶𝑛

𝛾∥𝐷𝜏∥∞ + 2−𝛾 ∥Δ𝜏∥∞ + ∥𝐷2𝜏∥∞

(cid:17)

.

Choosing 𝛾 =

(cid:16)

log ∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:17)

∨ 1 gives

∥ [W, 𝐿𝜏] ∥ ≤ ˜𝐶𝑛

(cid:18)(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

∥𝐷𝜏∥∞ + ∥𝐷2𝜏∥∞

(cid:19)

,

and the lemma is proved.

□

Theorem 20. Assume 𝜓 and its first and second order derivatives have decay in 𝑂 ((1 + |𝑥|)−𝑛−3)
and ∫

R𝑛 𝜓(𝑥) 𝑑𝑥 = 0. Then for every 𝜏 ∈ 𝐶2(R𝑛) with ∥𝐷𝜏∥∞ ≤ 1

2𝑛 , there exists 𝐶𝑚,𝑛 > 0 and

ˆ𝐶𝑚,𝑛 > 0 such that

∥𝑆𝑚

cont,2

𝑓 − 𝑆𝑚

cont,2

𝐿𝜏 𝑓 ∥2

L2 (R𝑚

+ ) ≤ 𝐶𝑚,𝑛𝐾2(𝜏) ∥ 𝑓 ∥2

2

.

and

with

∥𝑆𝑚

dyad,2

𝑓 − 𝑆𝑚

dyad,2

𝐿𝜏 𝑓 ∥2

ℓ2 (Z𝑚) ≤ ˆ𝐶𝑚,𝑛𝐾2(𝜏) ∥ 𝑓 ∥2

2

,

𝐾2(𝜏) = ∥𝐷𝜏∥2

∞ +

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 2

.

Proof. The proof is only provided for the continuous case. We have the following bound for some

𝐶𝑚:

∥𝑆𝑚

cont,2

𝑓 − 𝑆𝑚

cont,2

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ ) ≤ ∥ 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥L2 (R𝑚
+ )

+ ∥ 𝐴2𝑀 [W𝑉𝑚−1, 𝐿𝜏] 𝑓 ∥L2 (R𝑚
+ )

≤ ∥ 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥L2 (R𝑚
+ )

+ 𝐶2

𝑚 ∥ [W, 𝐿𝜏] ∥2

L2 (R𝑚

+ ×R𝑛)→L2 (R𝑛) ∥ 𝑓 ∥2

2

.

46

For the first term, we can mimic the dilation argument to get

| 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 | = |∥𝑔∥2 − ∥𝐿𝜏𝑔∥2| .

The difference is the term with the diffeomorphism. Let 𝑦 = 𝛾(𝑥) = 𝑥 − 𝜏(𝑥). Then it follows that

𝛾−1(𝑦) = 𝑥 and change of variables implies that

∥𝐿𝜏 𝑓 ∥2

2 =

∫

R𝑛

| 𝑓 (𝑥 − 𝜏(𝑥))|2 𝑑𝑥 =

∫

R𝑛

| 𝑓 (𝑦)|2

𝑑𝑦
| det(𝐼 − 𝐷𝜏(𝛾−1(𝑦)))|

.

We also have

Thus, we obtain

1 − 𝑛∥𝐷𝜏∥∞ ≤ | det(𝐼 − 𝐷𝜏(𝛾−1(𝑦)))| ≤ 1 + 𝑛∥𝐷𝜏∥∞.

∫

1
1 + 𝑛∥𝐷𝜏∥∞

| 𝑓 (𝑦)|2 𝑑𝑦 ≤ ∥𝐿𝜏 𝑓 ∥2

2 ≤

∥ 𝑓 ∥2

2 ≤ ∥𝐿𝜏 𝑓 ∥2

2 ≤

R𝑛
1
1 + 𝑛∥𝐷𝜏∥∞

1
1 − 𝑛∥𝐷𝜏∥∞
1
1 − 𝑛∥𝐷𝜏∥∞

∫

R𝑛

| 𝑓 (𝑦)|2 𝑑𝑦,

∥ 𝑓 ∥2
2

.

Since we have a bound on ∥𝐷𝜏∥∞, we see that

1
1 + 𝑛∥𝐷𝜏∥∞

=

1 − 𝑛∥𝐷𝜏∥∞
1 − 𝑛2∥𝐷𝜏∥2
∞

≥ 1 − 𝑛∥𝐷𝜏∥∞

since 1 > 1 − 𝑛2∥𝐷𝜏∥2

∞ > 0. Similarly,

1
1 − 𝑛∥𝐷𝜏∥∞

=

1 + 2𝑛∥𝐷𝜏∥∞
1 + 𝑛∥𝐷𝜏∥∞ − 2𝑛2∥𝐷𝜏∥2
∞

and

1 + 𝑛∥𝐷𝜏∥∞ − 2𝑛2∥𝐷𝜏∥2

∞ ≥ 1 + 𝑛∥𝐷𝜏∥∞ −

2𝑛2
2𝑛

∥𝐷𝜏∥∞ = 1

since ∥𝐷𝜏∥∞ ≤ 1

2𝑛 . It follows that

1
1−𝑛∥𝐷𝜏∥∞

≤ 1 + 2𝑛∥𝐷𝜏∥∞ and

(1 − 𝑛∥𝐷𝜏∥∞)1/2∥ 𝑓 ∥2 ≤ ∥𝐿𝜏 𝑓 ∥2 ≤ (1 + 2𝑛∥𝐷𝜏∥∞)1/2∥ 𝑓 ∥2.

Since 1 − 𝑛∥𝐷𝜏∥∞ < 1 and 1 + 2𝑛∥𝐷𝜏∥∞ > 1, Use the lower bound on ∥𝐿𝜏 𝑓 ∥2 to get

∥ 𝑓 ∥2 − ∥𝐿𝜏 𝑓 ∥2 = ∥ 𝑓 ∥2

(cid:16)

1 − (1 − 𝑛∥𝐷𝜏∥∞)1/2(cid:17)

≤ ∥ 𝑓 ∥2 (1 − (1 − 𝑛∥𝐷𝜏∥∞))

= 𝑛∥𝐷𝜏∥∞∥ 𝑓 ∥2.

47

and the upper bound to get

∥𝐿𝜏 𝑓 ∥2 − ∥ 𝑓 ∥2 = ∥ 𝑓 ∥2

(cid:16)

(1 + 2𝑛∥𝐷𝜏∥∞)1/2 − 1

(cid:17)

≤ ∥ 𝑓 ∥2 ((1 + 2𝑛∥𝐷𝜏∥∞) − 1)

= 2𝑛∥𝐷𝜏∥∞∥ 𝑓 ∥2.

Finally, we have

|∥ 𝑓 ∥2 − ∥𝐿𝜏 𝑓 ∥2| ≤ 2𝑛∥𝐷𝜏∥∞∥ 𝑓 ∥2

for any 𝑓 ∈ L2(R𝑛). Now we mimic the argument given for dilation stability to get

∥ 𝐴2𝑀W𝑉𝑚−1 𝑓 − 𝐴2𝑀 𝐿𝜏W𝑉𝑚−1 𝑓 ∥2

L2 (R𝑚

+ ) ≤ 𝐶 ∥𝐷𝜏∥2

∞∥ 𝑓 ∥2
2

for some constant 𝐶. For the second term, we have

𝐶2

𝑚 ∥ [W, 𝐿𝜏] ∥2

L2 (R𝑚

+ ×R𝑛)→L2 (R𝑛) ∥ 𝑓 ∥2

2 ≤ 𝐶′

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 2

∥ 𝑓 ∥2
2

for some constant 𝐶′. We now choose 𝐶𝑛,𝑚 = max{𝐶′, 𝐶} to get the desired bound.

□

2.4.2 Stability to Diffeomorphisms When 1 < 𝑞 < 2

Lemma 21. Let 𝛾(𝑧) = 𝑧 − 𝜏(𝑧), 𝑔(𝑧) = 𝑓 (𝛾(𝑧)), and

𝐾𝜆 (𝑥, 𝑧) = det(𝐷𝛾(𝑧))𝜓𝜆 (𝛾(𝑥) − 𝛾(𝑧)) − 𝜓𝜆 (𝑥 − 𝑧).

Additionally, define

𝑇𝜆𝑔(𝑥) =

∫

R𝑛

𝑔(𝑧)𝐾𝜆 (𝑥, 𝑧) 𝑑𝑧

and consider 𝑇 𝑔 : R𝑛 → L2(R+, 𝑑𝜆
X = L2(R+, 𝑑𝜆

𝜆𝑛+1 ),

𝜆𝑛+1 ) defined by 𝑇 𝑔(𝑥) = (𝑇𝜆𝑔(𝑥))𝜆∈R+

. Then for the Banach space

∥𝑇 𝑔∥2
X (R𝑛)
L2

≤ 𝐶𝑛,𝑚

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 2

∥ 𝑓 ∥2

for some constant 𝐶𝑛,𝑚 > 0.

48

Proof. Notice that

∥𝑇 𝑔∥2
𝑋 (R𝑛)
L2
∫ ∞
∫

=

=

=

=

R𝑛

∫

0
∫ ∞

R𝑛

∫

0
∫ ∞

R𝑛

∫

0
∫ ∞

R𝑛

0

|𝑇𝜆𝑔(𝑥)|2

𝑑𝜆
𝜆𝑛+1

𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

∫

R𝑛

∫

R𝑛

∫

R𝑛

𝐾𝜆 (𝑥, 𝑧)𝑔(𝑧) 𝑑𝑧

2 𝑑𝜆
𝜆𝑛+1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑥

𝑓 (𝛾(𝑧)) [det(𝐷𝛾(𝑧))𝜓𝜆 (𝛾(𝑥) − 𝛾(𝑧)) − 𝜓𝜆 (𝑥 − 𝑧)] 𝑑𝑧

2 𝑑𝜆
𝜆𝑛+1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑥

det(𝐷𝛾(𝑧)) 𝑓 (𝛾(𝑧))𝜓𝜆 (𝛾(𝑥) − 𝛾(𝑧)) 𝑑𝑧 −

∫

R𝑛

𝑓 (𝛾(𝑧))𝜓𝜆 (𝑥 − 𝑧) 𝑑𝑧

2 𝑑𝜆
𝜆𝑛+1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑥.

Using the change of variables 𝑢 = 𝛾(𝑧), we get

∥𝑇 𝑔∥2
𝑋 (R𝑛)
𝐿2

=

=

=

=

∫

∫ ∞

R𝑛

∫

0
∫ ∞

R𝑛
∫ ∞

0
∫

R𝑛

0
∫ ∞

0

|𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) (𝑥) − (𝐿𝜏 𝑓 ∗ 𝜓𝜆) (𝑥)|2

𝑑𝜆
𝜆𝑛+1

𝑑𝑥

| [W𝜆, 𝐿𝜏] 𝑓 (𝑥)|2

𝑑𝜆
𝜆𝑛+1

𝑑𝑥

| [W𝜆, 𝐿𝜏] 𝑓 (𝑥)|2 𝑑𝑥

𝑑𝜆
𝜆𝑛+1

∥ [W𝜆, 𝐿𝜏] 𝑓 ∥2
2

𝑑𝜆
𝜆𝑛+1

= ∥ [W, 𝐿𝜏] 𝑓 ∥2
(cid:18)

≤ 𝐶𝑛,𝑚

∥𝐷𝜏∥∞

L2 (R+×R𝑛)

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19)

∥ 𝑓 ∥2
2

,

where the last inequality follows from the 𝑞 = 2 case.

□

Lemma 22 ([39], Marcinkiewicz Interpolation). Let A and B be Banach spaces and let 𝑇 : A → B
𝑝1
A (R𝑛) with 0 < 𝑝0 < 𝑝1. Furthermore, if 𝑇

be a quasilinear operator defined on L

𝑝0
A (R𝑛) and L

satisfies

∥𝑇 𝑓 ∥L

𝑝𝑖 ,∞
B

(R𝑛) ≤ 𝑀𝑖 ∥ 𝑓 ∥L

𝑝𝑖
A (R𝑛)

for 𝑖 = 0, 1, then for all 𝑝 ∈ ( 𝑝0, 𝑝1),

∥𝑇 𝑓 ∥L

𝑝

B (R𝑛) ≤ 𝑁 𝑝 ∥ 𝑓 ∥L

𝑝

A (R𝑛),

where 𝑁 𝑝 only depends on 𝑀0, 𝑀1, and 𝑝.

49

Remark 7. Like with the scalar valued estimate, it can be shown that 𝑁 𝑝 = 𝜂𝑀 𝛿
0

𝑀 1−𝛿
1

, where

and

𝛿 =

𝑝0( 𝑝1 − 𝑝)
𝑝( 𝑝1 − 𝑝0)
𝑝0
𝑝





𝑝1 < ∞,

𝑝1 = ∞

(cid:19) 1/𝑝

𝜂 =

(cid:18)

(cid:18)

2

2






𝑝( 𝑝1 − 𝑝0)
( 𝑝 − 𝑝0) ( 𝑝1 − 𝑝)
(cid:19) 1/𝑝

𝑝0
𝑝 − 𝑝0

𝑝1 < ∞,

𝑝1 = ∞.

Lemma 23. Let 𝑇 be the operator defined in Lemma 21. Let 𝑞 ∈ (1, 2) and 𝑟 ∈ (1, 𝑞). Then 𝑇

satisfies

∥𝑇 𝑔∥L

𝑟 ,∞

X (R𝑛) ≤ 𝑀𝑟 ∥ 𝑓 ∥L𝑟 (R𝑛)

for some constant 𝑀𝑟 > 0, which is independent of ∥𝐷𝜏∥∞ and ∥𝐷2𝜏∥∞. Furthermore, 𝑇 also

satisfies

∥𝑇 𝑔∥2

L2,∞
X (R𝑛)

(cid:18)

≤ ˜𝐶𝑛

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

(cid:19) 2

+ ∥𝐷2𝜏∥∞

∥ 𝑓 ∥2

L2 (R𝑛)

for some constant ˜𝐶𝑛 > 0.

Proof. The second inequality obviously follows from strong boundedness of the operator, so we

will omit the proof. For the first inequality, the norm satisfies

∥𝑇 𝑔(𝑥)∥2

X =

=

∫ ∞

0
∫ ∞

∫

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)
0
∫ ∞

R𝑛

R𝑛
(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)

≤ 4

0

R𝑛

det(𝐷𝛾(𝑧)) 𝑓 (𝛾(𝑧))𝜓𝜆 (𝛾(𝑥) − 𝛾(𝑧)) 𝑑𝑧 −

∫

R𝑛

𝑓 (𝛾(𝑧))𝜓𝜆 (𝑥 − 𝑧) 𝑑𝑧

2 𝑑𝜆
𝜆𝑛+1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑓 (𝑧)𝜓𝜆 (𝛾(𝑥) − 𝑧) 𝑑𝑧 −

𝑓 (𝑧)𝜓𝜆 (𝛾(𝑥) − 𝑧) 𝑑𝑧

∫

R𝑛
(cid:12)
2 𝑑𝜆
(cid:12)
(cid:12)
𝜆2
(cid:12)

𝑓 (𝛾(𝑧))𝜓𝜆 (𝑥 − 𝑧) 𝑑𝑧

(cid:12)
2 𝑑𝜆
(cid:12)
(cid:12)
𝜆𝑛+1
(cid:12)

+ 4

∫ ∞

0

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∫

R𝑛

𝑓 (𝛾(𝑧))𝜓𝜆 (𝑥 − 𝑧) 𝑑𝑧

2 𝑑𝜆
𝜆𝑛+1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

= 4|(𝐺 𝑓 )(𝛾(𝑥))|2 + 4|𝐺 𝐿𝜏 𝑓 (𝑥)|2.

We see

∥𝑇 𝑔(𝑥)∥X ≤

√︃

4|(𝐺 𝑓 )(𝛾(𝑥))|2 + 4|𝐺 𝐿𝜏 𝑓 (𝑥)|2 ≤ 2|(𝐺 𝑓 ) (𝛾(𝑥))| + 2|𝐺 𝐿𝜏 𝑓 (𝑥)|.

50

For 𝛿 > 0, Chebyshev’s inequality implies that there exists 𝐴𝑟 such that

𝑚{∥𝑇 𝑔(𝑥)∥X > 𝛿} ≤ 𝑚{2|(𝐺 𝑓 ) (𝛾(𝑥))| + 2|𝐺 𝐿𝜏 𝑓 (𝑥)| > 𝛿}
𝐴𝑟
𝛿𝑟 (∥(𝐺 𝑓 ) (𝛾(·)) ∥𝑟

L𝑟 (R𝑛) + ∥𝐺 𝐿𝜏 𝑓 ∥𝑟

≤

L𝑟 (R𝑛)).

We want to now ensure that ∥(𝐺 𝑓 )(𝛾(·))∥𝑟

L𝑟 (R𝑛) can be bounded above by a constant multiple of

∥𝐺 𝑓 ∥𝑟

L𝑟 (R𝑛)

. Since 𝛾 is a diffeomorphism, we can use change of variables to get

∥(𝐺 𝑓 )(𝛾(·))∥𝑟

L𝑟 (R𝑛) =

=

|𝐺 𝑓 (𝛾(𝑥))|𝑟 𝑑𝑥

|𝐺 𝑓 (𝑢)|𝑟

𝑑𝑢
det (cid:2)(𝐷𝛾) (𝛾−1(𝑢))(cid:3)

∫

R𝑛

∫

R𝑛
∫

|𝐺 𝑓 (𝑥)|𝑟 𝑑𝑥

≤ 2

R𝑛
= 2∥𝐺 𝑓 ∥𝑟

.

L𝑟 (R𝑛)

By Theorem 5, we get

∥𝐺 𝐿𝜏 𝑓 ∥𝑟

L𝑟 (R𝑛) ≤ 𝐶𝑟 ∥𝐿𝜏 𝑓 ∥𝑟

L𝑟 (R𝑛) ≤ 2𝐶𝑟 ∥ 𝑓 ∥𝑟

L𝑟 (R𝑛)

for some constant 𝐶𝑟 dependent on 𝑟. Thus, we have

𝑚{∥𝑇 𝑔(𝑥) ∥X > 𝛿}1/𝑟 ≤

𝑀𝑟
𝛿

∥ 𝑓 ∥L𝑟 (R𝑛)

for some constant 𝑀𝑟 > 0.

□

Lemma 24. Fix 𝑟 = 1+𝑞

2 so that 𝑟 ∈ (1, 𝑞). For some constant 𝐶𝑛,𝑞 > 0, the operator 𝑇 defined in

Lemma 21 satisfies the estimate

∥𝑇 𝑔∥

𝑞

L

𝑞

X (R𝑛)

≤ 𝐶𝑛,𝑞𝜂𝑞 𝑀 𝑞𝛿
𝑟

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 𝑞(1−𝛿)

∥ 𝑓 ∥

𝑞
𝑞,

where 𝜂 and 𝛿 come from interpolation, and 𝑀𝑟 comes from the constant for weak boundedness in

Lemma 23.

Proof. Since 𝑇 is an integral operator, it is clear that is quasilinear. Using the L𝑟 (R𝑛) and L2(R𝑛)

estimates from the previous Lemma, we interpolate using Marcinkiewicz since ∥𝑔∥𝑟 ≤ 2∥ 𝑓 ∥𝑟 ≤

4∥𝑔∥𝑟.

□

51

Theorem 25. Let 1 < 𝑞 < 2. Assume 𝜓 and its first and second order derivatives have decay in
𝑂 ((1 + |𝑥|)−𝑛−3), and ∫

R𝑛 𝜓(𝑥) 𝑑𝑥 = 0. Then for every 𝜏 ∈ 𝐶2(R𝑛) with ∥𝐷𝜏∥∞ < 1

2𝑛 , there exists

𝐶𝑛,𝑞 > 0 such that

∥𝑆cont,𝑞 𝑓 − 𝑆cont,𝑞 𝐿𝜏 𝑓 ∥

𝑞
L2 (R+)

≤ 𝐶𝑛,𝑞𝐾𝑞 (𝜏) ∥ 𝑓 ∥

𝑞
𝑞

with

𝐾𝑞 (𝜏) = ∥𝐷𝜏∥

𝑞

∞ + 𝜂𝑞 𝑀 𝑞𝛿

𝑟

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 𝑞(1−𝛿)

.

Proof. We use the same notation as Theorem 16. Using a nearly identical argument to Corollary

17, we get

∥𝑆cont,𝑞 𝑓 − 𝑆cont,𝑞 𝐿𝜏 𝑓 ∥L2 (R+)

= ∥ 𝐴𝑞 𝑀W 𝑓 − 𝐴𝑞 𝑀W 𝐿𝜏 𝑓 ∥L2 (R+)

= ∥ 𝐴𝑞 𝑀W 𝑓 − 𝐴𝑞 𝑀 𝐿𝜏W 𝑓 + 𝐴𝑞 𝑀 𝐿𝜏𝑊 𝑓 − 𝐴𝑞 𝑀W 𝐿𝜏 𝑓 ∥L2 (R+)

≤ ∥ 𝐴𝑞 𝑀W 𝑓 − 𝐴𝑞 𝑀 𝐿𝜏W 𝑓 ∥L2 (R+) + ∥ 𝐴𝑞 𝑀 𝐿𝜏W 𝑓 − 𝐴𝑞 𝑀W 𝐿𝜏 𝑓 ∥L2 (R+)

≤ ∥( 𝐴𝑞 𝑀 − 𝐴𝑞 𝑀 𝐿𝜏)W 𝑓 ∥L2 (R+) + ∥ 𝐴𝑞 𝑀 [W, 𝐿𝜏] 𝑓 ∥L2 (R+).

The first term, ∥( 𝐴𝑞 𝑀 − 𝐴𝑞 𝑀 𝐿𝜏)W 𝑓 ∥L2 (R+), can be bounded using an argument identical to the
𝑞 = 2 case. In particular, we can prove that

(1 − 𝑛∥𝐷𝜏∥∞)∥ 𝑓 ∥𝑞 ≤ (1 − 𝑛∥𝐷𝜏∥∞)1/𝑞 ∥ 𝑓 ∥𝑞 ≤ ∥𝐿𝜏 𝑓 ∥𝑞

and

which means

For the other term,

∥𝐿𝜏 𝑓 ∥𝑞 ≤ (1 + 2𝑛∥𝐷𝜏∥∞)1/𝑞 ∥ 𝑓 ∥𝑞 ≤ (1 + 2𝑛∥𝐷𝜏∥∞) ∥ 𝑓 ∥𝑞,

∥( 𝐴𝑞 𝑀 − 𝐴𝑞 𝑀 𝐿𝜏)W 𝑓 ∥

𝑞
L2 (R+)

≤ 𝐶 ∥𝐷𝜏∥

𝑞
∞∥ 𝑓 ∥

𝑞
𝑞.

∥ 𝐴𝑞 𝑀 [W, 𝐿𝜏] 𝑓 ∥

𝑞
L2 (R+)

=

(cid:32)∫ ∞

(cid:20)∫

0

R𝑛

|(𝐿𝜏 𝑓 ∗ 𝜓𝜆) (𝑥) − 𝐿𝜏 ( 𝑓 ∗ 𝜓𝜆) (𝑥)|𝑞 𝑑𝑥

(cid:21) 2/𝑞 𝑑𝜆
𝜆𝑛+1

(cid:33) 𝑞/2

.

52

Now, expand convolution and then use change of variables to get

∥ 𝐴𝑞 𝑀 [W, 𝐿𝜏] 𝑓 ∥

𝑞
L2 (R+)

(cid:32)∫ ∞

(cid:20)∫

0
(cid:32)∫ ∞

0
(cid:32)∫ ∞

R𝑛

(cid:20)∫

R𝑛

(cid:20)∫

0

R𝑛

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∫

R𝑛

∫

R𝑛

𝑓 (𝛾(𝑧))(det(𝐷𝛾(𝑧))𝜓𝜆 (𝛾(𝑥) − 𝛾(𝑧)) − 𝜓𝜆 (𝑥 − 𝑧)) 𝑑𝑧

(cid:33) 𝑞/2

𝑞

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑥

(cid:21) 2/𝑞 𝑑𝜆
𝜆𝑛+1

𝑔(𝑧)𝐾𝜆 (𝑥, 𝑧) 𝑑𝑧

(cid:33) 𝑞/2

𝑞

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝑥

(cid:21) 2/𝑞 𝑑𝜆
𝜆𝑛+1

|𝑇𝜆𝑔(𝑥)|𝑞 𝑑𝑥

(cid:33) 𝑞/2

(cid:21) 2/𝑞 𝑑𝜆
𝜆𝑛+1

|𝑇𝜆𝑔(𝑥)|𝑞 𝑑𝜆
𝜆𝑛+1

(cid:21) 𝑞/2

|𝑇𝜆𝑔(𝑥)|2

(cid:21) 𝑞/2

𝑑𝜆
𝜆𝑛+1

𝑑𝑥

𝑑𝑥

∫

(cid:20)∫ ∞

R𝑛

∫

0
(cid:20)∫ ∞

0

R𝑛

∫

R𝑛

∥𝑇 𝑔(𝑥)∥

𝑞
L2 (cid:16)R+,

(cid:17)

𝑑𝜆
𝜆𝑛+1

𝑑𝑥

=

=

=

≤

=

=

= ∥𝑇 𝑔∥

𝑞

L

𝑞

X (R𝑛)
(cid:18)

≤ 𝐶𝑛𝜂𝑞 𝑀 𝑞𝛿
𝑟

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 𝑞(1−𝛿)

∥ 𝑓 ∥

𝑞
𝑞.

Thus, the proof is complete.

□

Corollary 26. Let 1 < 𝑞 < 2 . Assume 𝜓 and its first and second order derivatives have decay in
𝑂 ((1 + |𝑥|)−𝑛−3), and ∫

R𝑛 𝜓(𝑥) 𝑑𝑥 = 0. Then for every 𝜏 ∈ 𝐶2(R𝑛) with ∥𝐷𝜏∥∞ < 1

2𝑛 , there exist

constants 𝐶𝑛,𝑚, ˆ𝐶𝑛,𝑚 > 0 such that

∥𝑆𝑚

cont,𝑞 𝑓 − 𝑆𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

𝑞
L2 (R𝑚
+ )

≤ 𝐶𝑛,𝑚𝐾𝑞 (𝜏) ∥ 𝑓 ∥

𝑞
𝑞

and

∥𝑆𝑚

dyad,𝑞 𝑓 − 𝑆𝑚

dyad,𝑞 𝐿𝜏 𝑓 ∥

𝑞
ℓ2 (Z𝑚)

≤ ˆ𝐶𝑛,𝑚𝐾𝑞 (𝜏) ∥ 𝑓 ∥

𝑞
𝑞.

Remark 8. This bound is not exactly the same as the definition for stability to diffeomorphisms in
(cid:17)

[11], but the idea is similar. Since 𝑟 is fixed, so is 𝛿. It is easy to confirm that 𝛿 = 1

1+𝑞 ∈

(cid:16) 1
3

, 1
2

when using Marcinkiewicz interpolation in Lemma 24, so

𝐶𝑛,𝑞𝜂𝑞 𝑀 𝑞𝛿
𝑟

(cid:18)

∥𝐷𝜏∥∞

(cid:18)

log

∥Δ𝜏∥∞
∥𝐷𝜏∥∞

(cid:19)

∨ 1

+ ∥𝐷2𝜏∥∞

(cid:19) 𝑞(1−𝛿)

→ 0

53

when ∥𝐷𝜏∥∞ → 0 and ∥𝐷2𝜏∥∞ → 0.

2.5 Equivariance and Invariance to Rotations

We now consider adding group actions to our scattering transform and prove invariance to

rotations. Let SO(𝑛) be the group of 𝑛 × 𝑛 rotation matrices. Since SO(𝑛) is a compact Lie group,

we can define a Haar measure, say 𝜇, with 𝜇(SO(𝑛)) < ∞. We say that 𝑓 ∈ L2(SO(𝑛)) if and only
if 𝑓 is 𝜇-measurable and ∫

| 𝑓 (𝑟)|2 𝑑𝜇(𝑟) < ∞.

SO(𝑛)

2.5.1 Rotation Equivariant Representations

Let 𝜓 : R𝑛 → R be a wavelet. Define

𝜓𝜆,𝑅 (𝑥) = 𝜆−𝑛/2𝜓(𝜆−1𝑅−1𝑥),

where 𝑅 ∈ SO(𝑛) is a 𝑛 × 𝑛 rotation matrix. The continuous and dyadic wavelet transforms of 𝑓

are given by

WRot 𝑓 := { 𝑓 ∗ 𝜓𝜆,𝑅 (𝑥) : 𝑥 ∈ R𝑛, 𝜆 ∈ (0, ∞), 𝑅 ∈ SO(𝑛)},

𝑊Rot 𝑓 := { 𝑓 ∗ 𝜓 𝑗,𝑅 (𝑥) : 𝑥 ∈ R𝑛, 𝑗 ∈ Z, 𝑅 ∈ SO(𝑛)}.

We will first consider a translation invariant and rotation equivariant formulation of continuous and

dyadic one-layer scattering using

𝔖cont,𝑞 𝑓 (𝜆, 𝑅) := ∥ 𝑓 ∗ 𝜓𝜆,𝑅 ∥𝑞,

𝔖dyad,𝑞 𝑓 ( 𝑗, 𝑅) := ∥ 𝑓 ∗ 𝜓 𝑗,𝑅 ∥𝑞.

The translation invariance of our representation follows from translation invariance of the norm.

For rotation equivariance, notice that if 𝑓 ˜𝑅 (𝑥) := 𝑓 ( ˜𝑅−1𝑥), then we have

𝔖cont,𝑞 𝑓 ˜𝑅 (𝜆, 𝑅) = 𝔖cont,𝑞 𝑓 (𝜆, ˜𝑅−1𝑅),

𝔖dyad,𝑞 𝑓 ˜𝑅 ( 𝑗, 𝑅) = 𝔖dyad,𝑞 𝑓 ( 𝑗, ˜𝑅−1𝑅).

Now suppose we have 𝑚 layers again. Then we define our 𝑚 layer transforms by

𝔖𝑚

cont,𝑞 𝑓 (𝜆1, . . . , 𝜆𝑚, 𝑅1, . . . , 𝑅𝑚) := ∥| 𝑓 ∗ 𝜓𝜆1,𝑅1 | ∗ . . . | ∗ 𝜓𝜆𝑚,𝑅𝑚 ∥𝑞,

𝔖𝑚

dyad,𝑞 𝑓 ( 𝑗1, . . . , 𝑗𝑚, 𝑅1, . . . , 𝑅𝑚) := ∥| 𝑓 ∗ 𝜓 𝑗1,𝑅1 | ∗ . . . | ∗ 𝜓 𝑗𝑚,𝑅𝑚 ∥𝑞.

54

and rotation equivariance implies

𝔖𝑚

cont,𝑞 𝑓 ˜𝑅 (𝜆1, . . . , 𝜆𝑚, 𝑅1, . . . , 𝑅𝑚) = 𝔖𝑚
dyad,𝑞 𝑓 ˜𝑅 ( 𝑗1, . . . , 𝑗𝑚, 𝑅1, . . . , 𝑅𝑚) = 𝔖𝑚
𝔖𝑚

cont,𝑞 𝑓 (𝜆1, . . . , 𝜆𝑚, ˜𝑅−1𝑅1, . . . , ˜𝑅−1𝑅𝑚),

dyad,𝑞 𝑓 ( 𝑗1, . . . , 𝑗𝑚, ˜𝑅−1𝑅1, . . . , ˜𝑅−1𝑅𝑚).

The norm we will use is similar to our previous formulations. Denote the scattering norm for

the continuous transform as ∥𝔖𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚

+ )×SO(𝑛)𝑚, which is defined as

(cid:32)∫ ∞

∫

∫ ∞

∫

· · ·

0

SO(𝑛)

0

SO(𝑛)

∥| 𝑓 ∗ 𝜓 𝑗1,𝑅1 | ∗ . . . | ∗ 𝜓 𝑗𝑚,𝑅𝑚 ∥2

𝑞𝑑𝜇1(𝑅1)

𝑑𝜆1
𝜆𝑛+1
1

. . . 𝑑𝜇𝑚 (𝑅𝑛)

(cid:33) 𝑞/2

.

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

For the dyadic transform, we denote the norm using ∥𝔖𝑚

dyad,𝑞 𝑓 ∥

𝑞
ℓ2 (Z𝑚)×SO(𝑛)𝑚, which is given by

(cid:32)

∫

∑︁

∫

∑︁

· · ·

𝑗𝑚∈Z

SO(𝑛)

𝑗1∈Z

SO(𝑛)

∥| 𝑓 ∗ 𝜓 𝑗1,𝑅1 | ∗ . . . | ∗ 𝜓 𝑗𝑚,𝑅𝑚 ∥2

𝑞𝑑𝜇1(𝑅1) . . . 𝑑𝜇𝑚 (𝑅𝑛)

(cid:33) 𝑞/2

.

We will start by proving that these formulations of the scattering transform are well defined, and

prove properties about stability to diffeomorphisms like in previous chapters.

Lemma 27. Let 𝜓 be a wavelet that satisfies properties (2.4) and (2.5).

• If 1 < 𝑞 ≤ 2, we have 𝔖𝑚

cont,𝑞 : L𝑞 (R𝑛) → L2(R𝑚

+ ) × SO(𝑛)𝑚 and 𝔖𝑚

dyad,𝑞 : L𝑞 (R𝑛) →

ℓ2(Z𝑚) × SO(𝑛)𝑚.

• If 𝑞 = 1 and one of the following holds:

– 𝑛 = 1 and 𝜓 is complex analytic,

– 𝑛 ≥ 2 and 𝜓 satisfies the conditions of Lemma 9,

then 𝔖𝑚

cont,1 : L1(R𝑛) → L2(R𝑚

+ ) × SO(𝑛)𝑚 and 𝔖𝑚

dyad,1 : L1(R𝑛) → ℓ2(Z𝑚) × SO(𝑛)𝑚.

• If 𝜓 is also a Littlewood-Paley wavelet, we have

∥𝔖𝑚

cont,2

𝑓 ∥2

∥𝔖𝑚

dyad,𝑞 𝑓 ∥2

+ )×SO(𝑛)𝑚 = 𝜇(SO(𝑛))𝑚𝐶𝑚
L2 (R𝑚
ℓ2 (Z𝑚)×SO(𝑛)𝑚 = 𝜇(SO(𝑛))𝑚 ˆ𝐶𝑚

𝜓 ∥ 𝑓 ∥2
2

𝜓 ∥ 𝑓 ∥2
2

,

.

Proof. We prove the first and third claim. The second claim is almost identical to the first claim, so

the proof will be omitted for brevity. Note that we will only provide arguments for the continuous

55

scattering transform since the proofs for the dyadic transform are very similar. By Fubini Theorem

and boundedness of the 𝑚-layer scattering transform, there exists a constant 𝐶𝑞 > 0, which is

dependent on 𝑞, such that

∥𝔖𝑚

cont,𝑞 𝑓 ∥
(cid:34)∫ ∞

∫

𝑞
L2 (R𝑚

+ )×SO(𝑛)𝑚
∫ ∞

∫

· · ·

=

≤

0
(cid:20)∫

SO(𝑛)
∫

· · ·

0

SO(𝑛)

(𝐶𝑚𝑞

𝑞 ∥ 𝑓 ∥

SO(𝑛)

SO(𝑛)
𝑞 𝜇(SO(𝑛))𝑚𝑞/2∥ 𝑓 ∥

𝑞
𝑞

= 𝐶𝑚𝑞

∥| 𝑓 ∗ 𝜓𝜆1,𝑅1 | ∗ . . . | ∗ 𝜓𝜆𝑚,𝑅𝑚 ∥2

𝑞𝑑𝜇(𝑅𝑚)

𝑞
𝑞)2/𝑞 𝑑𝜇(𝑅1) · · · 𝑑𝜇(𝑅𝑚)

(cid:21) 𝑞/2

𝑑𝜆1
𝜆𝑛+1
1

· · · 𝑑𝜇(𝑅1)

(cid:35) 𝑞/2

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

because each 𝜓𝜆𝑖,𝑅𝑖 is still a wavelet with sufficient decay even if the rotation is applied. For the

third claim, we see that

∥𝔖𝑚
cont,2
∫

=

𝑓 ∥2

L2 (R𝑚
∫

+ )×SO(𝑛)𝑚
𝐶𝑚

· · ·

SO(𝑛)

SO(𝑛)
= 𝜇(SO(𝑛))𝑚𝐶𝑚

𝜓 ∥ 𝑓 ∥2
2

𝜓 ∥ 𝑓 ∥2
2

𝑑𝜇(𝑅1) · · · 𝑑𝜇(𝑅𝑚)

.

□

Theorem 28. Assume |𝑐| < 1

2𝑛 . Let 𝜏(𝑥) = 𝑐𝑥 and 𝐿𝜏 𝑓 (𝑥) = 𝑓 ((1 − 𝑐)𝑥). Suppose that 𝜓 is

a wavelet that satisfies the conditions of Lemma 14. Then there exist constants ˜𝐾𝑛,𝑚,𝑞 and ˜𝐾′

𝑛,𝑚,𝑞

dependent only on 𝑛, 𝑚, and 𝑞 such that

∥𝔖𝑚

cont,𝑞 𝑓 − 𝔖𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

𝑞
L2 (R𝑚

+ )×SO(𝑛)𝑚 ≤ |𝑐|𝑞 · ˜𝐾𝑛,𝑚,𝑞 ∥ 𝑓 ∥

𝑞
𝑞

and

∥𝔖𝑚

dyad,𝑞 𝑓 − 𝔖𝑚

dyad,𝑞 𝐿𝜏 𝑓 ∥

𝑞

ℓ2 (Z𝑚)×SO(𝑛)𝑚 ≤ |𝑐|𝑞 · ˜𝐾′

𝑛,𝑚,𝑞 ∥ 𝑓 ∥

𝑞
𝑞.

Alternatively, if one of the following holds:

• 𝑛 = 1, 𝜓 is complex analytic and satisfies the conditions of Lemma 14,

• 𝑛 ≥ 2 and 𝜓 satisfies the conditions of Lemma 9,

56

there exist ˜𝐻𝑚,𝑛 and ˜𝐻′

𝑚,𝑛 such that

∥𝔖𝑚

cont,1

𝑓 − 𝔖𝑚

cont,1

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ )×SO(𝑛)𝑚 ≤ |𝑐| · ˜𝐻𝑚,𝑛 ∥ 𝑓 ∥H1 (R𝑛).

and

∥𝔖𝑚

dyad,1

𝑓 − 𝔖𝑚

dyad,1

𝐿𝜏 𝑓 ∥ℓ2 (Z𝑚)×SO(𝑛)𝑚 ≤ |𝑐| · ˜𝐻′

𝑚,𝑛 ∥ 𝑓 ∥H1 (R𝑛)

Theorem 29. Let 𝜏 ∈ 𝐶2(R𝑛), and let 𝐿𝜏 𝑓 (𝑥) = 𝑓 (𝑥 − 𝜏(𝑥)). Suppose that 𝜓 is a wavelet such

that the wavelet and all its first and second partial derivatives have 𝑂 ((1 + |𝑥|)−𝑛−3) decay. When

𝑞 ∈ (1, 2), there exists a constant 𝐶𝑛,𝑚,𝑞 dependent on 𝜇(SO(𝑛)), 𝑛, 𝑚, and 𝑞 such that

∥𝔖𝑚

cont,𝑞 𝑓 − 𝔖𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

∥𝔖𝑚

dyad,𝑞 𝑓 − 𝔖𝑚
𝑓 − 𝔖𝑚

cont,2

∥𝔖𝑚

dyad,𝑞 𝐿𝜏 𝑓 ∥

𝐿𝜏 𝑓 ∥2

cont,2

∥𝔖𝑚

cont,2

𝑓 − 𝔖𝑚

cont,2

𝐿𝜏 𝑓 ∥2

𝑞
𝑞,

𝑞
𝑞,

𝑞

𝑞
+ )×SO(𝑛)𝑚 ≤ 𝐶𝑛,𝑚,𝑞𝐾𝑞 (𝜏) ∥ 𝑓 ∥
L2 (R𝑚
ℓ2 (Z𝑚)×SO(𝑛)𝑚 ≤ ˜𝐶𝑛,𝑚,𝑞𝐾𝑞 (𝜏) ∥ 𝑓 ∥
,
+ )×SO(𝑛)𝑚 ≤ 𝐶𝑛,𝑚𝐾2(𝜏) ∥ 𝑓 ∥2
+ )×SO(𝑛)𝑚 ≤ 𝐶𝑛,𝑚𝐾2(𝜏) ∥ 𝑓 ∥2

L2 (R𝑚

L2 (R𝑚

2

2

.

2.5.2 Rotation Invariant Representations

The representation before was rotation equivariant, but in some tasks, we would rather have

rotation invariance. In [11], the authors choose to integrate over each group action in a group of

transformations. However, this will remove the information the relative angles between each action

if we have multiple layers in our transform.

In the case of one layer, since there is only one angle, we use a similar formulation to [11] and

define continuous and dyadic scattering transforms for rotation invariance as

𝒮cont,𝑞 𝑓 (𝜆) =

𝒮dyad,𝑞 𝑓 ( 𝑗) =

∫

SO(𝑛)

∫

SO(𝑛)

∥ 𝑓 ∗ 𝜓𝜆,𝑅 ∥

𝑞
L𝑞 (R𝑛)

𝑑𝜇(𝑅),

∥ 𝑓 ∗ 𝜓 𝑗,𝑅 ∥

𝑞
L𝑞 (R𝑛)

𝑑𝜇(𝑅).

The corresponding norms are given by

∥𝒮cont,𝑞 𝑓 ∥

𝑞

L2 (R+) :=

(cid:34)∫ ∞

(cid:20)∫

0

SO(𝑛)

∥ 𝑓 ∗ 𝜓𝜆,𝑅 ∥𝑞 𝜇(𝑅)

(cid:21) 2/𝑞 𝑑𝜆
𝜆𝑛+1

(cid:35) 𝑞/2

,

∥𝒮dyad,𝑞 𝑓 ∥

𝑞
ℓ2 (Z) :=

(cid:34)

∑︁

(cid:20)∫

𝑗 ∈Z

SO(𝑛)

∥ 𝑓 ∗ 𝜓 𝑗,𝑅 ∥𝑞 𝜇(𝑅)

(cid:21) 2/𝑞(cid:35) 𝑞/2

.

57

Now we generalize to the case where 𝑚 ≥ 2. Let 𝑅1, . . . , 𝑅𝑚 ∈ SO(𝑛). Define

𝒮𝑚
cont,𝑞 𝑓 (𝜆1, . . . , 𝜆𝑚, 𝑅2, . . . , 𝑅𝑚) :=

𝒮𝑚
dyad,𝑞 𝑓 ( 𝑗1, . . . , 𝑗𝑚, 𝑅2, . . . , 𝑅𝑚) :=

∫

SO(𝑛)

∫

SO(𝑛)

∥| 𝑓 ∗ 𝜓𝜆1,𝑅2𝑅1 | ∗ · · · ∗ |𝜓𝜆𝑚,𝑅𝑚𝑅1 ∥2

𝑞 𝑑𝜇(𝑅1),

∥| 𝑓 ∗ 𝜓 𝑗1,𝑅2𝑅1 | ∗ . . . | ∗ 𝜓 𝑗𝑚,𝑅𝑚𝑅1 ∥2

𝑞 𝑑𝜇(𝑅1).

The norm for the continuous transform, the norm ∥𝒮𝑚

cont,𝑞 𝑓 ∥

𝑞
L2 (R𝑚

+ )×SO(𝑛)𝑚−1, is given by

(cid:32)∫ ∞

∫

∫ ∞

∫

∫ ∞

· · ·

0

SO(𝑛)

0

SO(𝑛)

0

𝒮𝑚
cont,𝑞 𝑓

𝑑𝜆1
𝜆𝑛+1
1

𝑑𝜇2(𝑅2)

𝑑𝜆2
𝜆𝑛+1
2

. . . 𝑑𝜇𝑚 (𝑅𝑚)

(cid:33) 𝑞/2

,

𝑑𝜆𝑚
𝜆𝑛+1
𝑚

where we use the shorthand notation

cont,𝑞 𝑓 := 𝒮𝑚
𝒮𝑚

cont,𝑞 𝑓 (𝜆1, . . . , 𝜆𝑚, 𝑅2, . . . , 𝑅𝑚)

dyad,𝑞 𝑓 := 𝒮𝑚
𝒮𝑚

dyad,𝑞 𝑓 (𝜆1, . . . , 𝜆𝑚, 𝑅2, . . . , 𝑅𝑚)

and

for brevity.

For the dyadic transform, the norm ∥𝒮𝑚

dyad,𝑞 𝑓 ∥

𝑞
ℓ2 (Z)×SO(𝑛)𝑚−1 is given by

(cid:32)

∫

∑︁

∫

∑︁

· · ·

𝑗𝑚∈Z

SO(𝑛)

𝑗2∈Z

SO(𝑛)

∑︁

𝑗1∈Z

𝒮𝑚
dyad,𝑞 𝑓 𝑑𝜇1(𝑅1) 𝑑𝜇2(𝑅2) . . . 𝑑𝜇𝑚 (𝑅𝑚)

(cid:33) 𝑞/2

.

Like before, we will discuss the well-definedness and stability of these operators to diffeomorphisms.

The proofs will be omitted since they follow directly from the previous sections with minor

modifications.

Lemma 30. Let 𝜓 be a wavelet that satisfies properties (2.4) and (2.5).

• If 1 < 𝑞 ≤ 2, we have 𝒮𝑚

cont,𝑞 : L𝑞 (R𝑛) → L2(R𝑚

+ ) × SO(𝑛)𝑚−1 and 𝒮𝑚

dyad,𝑞 : L𝑞 (R𝑛) →

ℓ2(Z𝑚) × SO(𝑛)𝑚−1.

• If 𝑞 = 1 and one of the following holds:

– 𝑛 = 1 and 𝜓 is complex analytic,

– 𝑛 ≥ 2 and 𝜓 satisfies the conditions of Lemma 9,
+ ) × SO(𝑛)𝑚−1 and 𝒮𝑚

cont,1 : L1(R𝑛) → L2(R𝑚

then 𝒮𝑚

dyad,1 : L1(R𝑛) → ℓ2(Z𝑚) × SO(𝑛)𝑚−1.

58

• If 𝑞 = 2 and 𝜓 is also a littlewood paley wavelet, we have ∥𝒮𝑚

𝑓 ∥ℓ1 (Z𝑚)×SO(𝑛)𝑚−1 =

𝜇(SO(𝑛))𝑚−1𝐶𝑚

𝜓 ∥ 𝑓 ∥2

2 and ∥𝒮𝑚

cont,2

𝑓 ∥L1 (R𝑚

dyad,2
+ )×SO(𝑛)𝑚−1 = 𝜇(SO(𝑛))𝑚−1 ˆ𝐶𝑚

𝜓 ∥ 𝑓 ∥2
2.

Theorem 31. Assume |𝑐| < 1

2𝑛 and 1 < 𝑞 < 2. Let 𝜏(𝑥) = 𝑐𝑥 and let 𝐿𝜏 𝑓 (𝑥) = 𝑓 ((1 − 𝑐)𝑥).
Suppose that 𝜓 is a wavelet that satisfies the conditions of Lemma 14. Then there exist constants

ˆ𝐾𝑛,𝑚,𝑞 and ˆ𝐾′

𝑛,𝑚,𝑞 dependent only on 𝑛, 𝑚, and 𝑞 such that

∥𝒮𝑚

cont,𝑞 𝑓 − 𝒮𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

𝑞
L2 (R𝑚

+ )×SO(𝑛)𝑚−1 ≤ |𝑐|𝑞 · ˆ𝐾𝑛,𝑚,𝑞 ∥ 𝑓 ∥

𝑞
𝑞

and

∥𝒮𝑚

dyad,𝑞 𝑓 − 𝒮𝑚

dyad,𝑞 𝐿𝜏 𝑓 ∥

𝑞

ℓ2 (Z𝑚)×SO(𝑛)𝑚−1 ≤ |𝑐|𝑞 · ˆ𝐾′

𝑛,𝑚,𝑞 ∥ 𝑓 ∥

𝑞
𝑞.

Additionally, if 𝑞 = 1 and one of the following holds:

• 𝑛 = 1, 𝜓 is complex analytic and satisfies the conditions of Lemma 14,

• 𝑛 ≥ 2 and 𝜓 satisfies the conditions of Lemma 9,

there exist ˆ𝐻𝑚,𝑛 and ˆ𝐻′

𝑚,𝑛 such that

∥𝒮𝑚

cont,1

𝑓 − 𝒮𝑚

cont,1

𝐿𝜏 𝑓 ∥L2 (R𝑚

+ )×SO(𝑛)𝑚−1 ≤ |𝑐| · ˆ𝐻𝑚,𝑛 ∥ 𝑓 ∥H1 (R𝑛)

and

∥𝒮𝑚

dyad,1

𝑓 − 𝒮𝑚

dyad,1

𝐿𝜏 𝑓 ∥ℓ2 (Z𝑚)×SO(𝑛)𝑚−1 ≤ |𝑐| · ˆ𝐻′

𝑚,𝑛 ∥ 𝑓 ∥H1 (R𝑛).

Theorem 32. Let 𝜏 ∈ 𝐶2(R𝑛) and define 𝐿𝜏 𝑓 (𝑥) = 𝑓 (𝑥 − 𝜏(𝑥)) with ∥𝐷𝜏∥∞ < 1
2𝑛 . Suppose that 𝜓
is a wavelet such that the wavelet and all its first and second partial derivatives have 𝑂 ((1+ |𝑥|)−𝑛−3)

decay. For 𝑞 ∈ (1, 2], there exist constants 𝐶𝑚,𝑛, ˆ𝐶𝑚,𝑛, 𝐶𝑚,𝑛,𝑞, and ˆ𝐶𝑚,𝑛,𝑞 such that

∥𝒮𝑚

cont,2

𝑓 − 𝒮𝑚

cont,2

𝐿𝜏 𝑓 ∥2

∥𝒮𝑚

dyad,2

𝑓 − 𝒮𝑚

dyad,2

𝐿𝜏 𝑓 ∥2

∥𝒮cont,𝑞 𝑓 − 𝒮𝑚

cont,𝑞 𝐿𝜏 𝑓 ∥

∥𝒮𝑚

dyad,𝑞 𝑓 − 𝒮𝑚

dyad,𝑞 𝐿𝜏 𝑓 ∥

,

,

2

+ )×SO(𝑛)𝑚−1 ≤ 𝐶𝑚,𝑛 𝐾2(𝜏) ∥ 𝑓 ∥2
L2 (R𝑚
ℓ2 (Z𝑚)×SO(𝑛)𝑚−1 ≤ ˆ𝐶𝑚,𝑛 𝐾2(𝜏) ∥ 𝑓 ∥2
𝑞
+ )×SO(𝑛)𝑚−1 ≤ 𝐶𝑚,𝑛,𝑞𝐾𝑞 (𝜏) ∥ 𝑓 ∥
L2 (R𝑚
ℓ2 (Z𝑚)×SO(𝑛)𝑚−1 ≤ ˆ𝐶𝑚,𝑛,𝑞𝐾𝑞 (𝜏) ∥ 𝑓 ∥

𝑞

2

𝑞
𝑞,

𝑞
𝑞

59

CHAPTER 3

EXPECTED SCATTERING TRANSFORMS

3.1 Background

Generalizing to stochastic processes, one can also consider scattering moments [11, 15], which

have similar desirable properties as the nonwindowed scattering transform; other tangential works

include [43, 44]. For the modeling of objects such as audio and image textures, one can think of

them as realizations of highly non-Gaussian processes [15].

In the particular case of audio/image synthesis in particular, one would like generate a texture

with the same statistical properties without generating a repetition of the texture. Equivariant

features are more likely to lead to repetitions in textures. Thus, it is sensible to get a small number

of rich descriptors that are translation invariant (e.g. using a realization of a process and calculating

the nonwindowed scattering transform).

In practice, instead of calculating an expectation, one

takes an average of multiple realizations. Applications further applications include cosmology

[45]. The main idea in all these applications is that the nonwindowed scattering transform has

desirable mathematical properties and provides a small number of relevant descriptors for high

dimensional, complicated data.

3.2 Wavelet Transforms for Stochastic Processes

Let 𝑋 be a real valued stationary stochastic process with finite second moment. Also, let 𝜓 be

a wavelet. As a reminder, let 𝐺 be a finite rotation group, and 𝐺+ be the quotient of 𝐺 with the set

{−1, 1}, and let

Λ = {(2 𝑗 , 𝑟) :

𝑗 ∈ Z, 𝑟 ∈ 𝐺+}.

For all 𝜆 ∈ Λ, dilations of the wavelet are given by

𝜓𝜆 (𝑢) = 2−𝑛 𝑗 𝜓(2− 𝑗𝑟 −1𝑢),

and we define the wavelet transform of 𝑋 at scale 2 𝑗 as

𝑋 ∗ 𝜓 𝑗 (𝑡) =

∫

R𝑛

𝑋 (𝑢)𝜓 𝑗 (𝑡 − 𝑢) 𝑑𝑢.

60

(3.1)

(3.2)

The dyadic wavelet transform is given by

𝑊 𝑋 = {𝑋 ∗ 𝜓𝜆}𝜆∈Λ.

(3.3)

We say that is 𝜓 a littlewood paley wavelet if 𝜓 satisfies the following admissibility condition:

| ˆ𝜓𝜆 (𝜔)|2 =

∑︁

𝜆∈Λ

∑︁

∑︁

𝑟∈𝐺+

𝑗 ∈Z

| ˆ𝜓(2 𝑗𝑟 −1𝜔)|2 = 𝐶𝜓,

∀𝜔 ≠ 0.

(3.4)

For any littlewood paley wavelet, we have the following relation between the variance 𝜎2(𝑋) and

the energy of the wavelet transform:

where

∑︁

𝜆∈Λ

𝛽 =





E[|𝑋 ∗ 𝜓𝜆|2] = 𝛽𝐶𝜓 𝜎2(𝑋),

(3.5)

1/2

if 𝜓 is real valued,

1

if 𝜓 is complex valued.

3.3 Scattering Moments and the Expected Scattering Transform

Following [11], first order scattering moments are defined as

𝑆1𝑋 (𝜆) = E [|𝑋 ∗ 𝜓𝜆|] ,

∀𝜆 ∈ Λ.

(3.6)

Scattering moments for 𝑚 > 1 are an iterative application of a wavelet transform followed by a

modulus, which is given by:

𝑆𝑚
1

𝑋 (𝜆1, . . . , 𝜆𝑚) = E (cid:2)||𝑋 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑚 |(cid:3) ,

∀(𝜆1, . . . , 𝜆𝑚) ∈ Λ𝑚.

(3.7)

The expected scattering transform is the set of all scattering moments:

𝑆1𝑋 = {𝑆𝑚
1

𝑋 (𝜆1, . . . , 𝜆𝑚) : ∀(𝜆1, . . . , 𝜆𝑚) ∈ Λ𝑚, ∀𝑚 ∈ N}

with norm

∞
∑︁

∑︁

∥𝑆1𝑋 ∥2 =

𝑚=1

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚
1

𝑋 (𝜆1, . . . , 𝜆𝑚)|2.

(3.8)

(3.9)

Additionally, suppose that 𝑌 is also a stochastic process with finite second moment. The scattering

distance is given by

∥𝑆1𝑋 − 𝑆1𝑌 ∥2 =

∞
∑︁

∑︁

𝑚=1

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚
1

𝑋 (𝜆1, . . . , 𝜆𝑚) − 𝑆𝑚
1

𝑌 (𝜆1, . . . , 𝜆𝑚)|2

(3.10)

61

3.4 The Expected Scattering Transform When 𝑞 = 2

Generalizing the norms above, we begin by defining the expected scattering transform and

scattering norm when 𝑞 = 2. The expected scattering transform is the set of all scattering moments:

𝑆2𝑋 = {𝑆𝑚
2

𝑋 (𝜆1, . . . , 𝜆𝑚) : ∀(𝜆1, . . . , 𝜆𝑚) ∈ Λ𝑚, ∀𝑚 ∈ N}

with norm

and scattering distance

∥𝑆2𝑋 ∥2

2 =

∞
∑︁

∑︁

𝑚=1

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚
2

𝑋 (𝜆1, . . . , 𝜆𝑚)|2

(3.11)

(3.12)

∥𝑆2𝑋 − 𝑆2𝑌 ∥2

2 =

∞
∑︁

∑︁

𝑚=1

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚
2

𝑋 (𝜆1, . . . , 𝜆𝑚) − 𝑆𝑚
2

𝑌 (𝜆1, . . . , 𝜆𝑚)|2

(3.13)

3.4.1 General Properties

Lemma 33. Suppose 𝜓 is a littlewood paley wavelet. Then we have the following bound:

∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚
2

𝑋 (𝜆1, . . . , 𝜆𝑚)|2 = 𝛽𝑚𝐶𝑚

𝜓 𝜎2(𝑋) ≤ 𝛽𝑚𝐶𝑚

𝜓 E[𝑋 2].

Proof. Without a loss of generality, assume that 𝜓 is complex and remove 𝛽 from all the proofs.

We proceed by induction. The base case follows directly from (3.5) since Thus, we have

∥𝑆2𝑋 ∥2

ℓ2 (Z) = 𝐶𝜓𝜎2(𝑋) ≤ 𝐶𝜓E[𝑋 2].

Now assume that for some 𝑘 ∈ N,

∑︁

|𝑆𝑘
2

(𝜆1,...,𝜆𝑘)∈Λ𝑘

𝑋 (𝜆1, . . . , 𝜆𝑘 )|2 = 𝐶 𝑘

𝜓𝜎2(𝑋) ≤ 𝐶 𝑘

𝜓E[𝑋 2].

Define the random variable 𝑌𝑘 = ||𝑋 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑘 |, which is clearly stationary since the

the modulus operator and wavelet transform both preserve stationarity of a stochastic process. It

62

follows that we can write

∑︁

|𝑆𝑘+1
2

𝑋 (𝜆1, . . . , 𝜆𝑘+1)|2 =

∑︁

∑︁

E[|𝑌𝑘 ∗ 𝜓 𝑗𝑘+1 |]2

(𝜆1,...,𝜆𝑘+1)∈Λ𝑘+1

(𝜆1,...,𝜆𝑘)∈Λ𝑘
∑︁

= 𝐶𝜓

𝜆𝑘+1∈Λ

𝜎2(𝑌𝑘 )

≤ 𝐶𝜓

( 𝑗1,..., 𝑗𝑘)∈Z𝑘
∑︁

( 𝑗1,..., 𝑗𝑘)∈Z𝑘

≤ 𝐶 𝑘+1

𝜓 E[𝑋 2].

E[𝑌 2
𝑘 ]

We first begin by proving that our expected scattering transform with 𝑞 = 2 is a nonexpansive

operator.

Theorem 34 (Nonexpansive Operator). Suppose 𝜓 is a littlewood paley wavelet with 𝛽𝐶𝜓 ≤ 1
2.

□

Then ∥𝑆2𝑋 − 𝑆2𝑌 ∥2

2 ≤ E[|𝑋 − 𝑌 |2] and ∥𝑆2𝑋 ∥2

2 ≤ E[𝑋 2].

Proof. For notational simplicity, define

𝑋𝑘 = ||𝑋 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑘 |,

𝑌𝑘 = ||𝑌 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑘 |.

63

We begin by applying Minkowski’s inequality and (3.5) repeatedly to get

∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚
2

𝑋 (𝜆1, . . . , 𝜆𝑚) − 𝑆𝑚
2

𝑌 (𝜆1, . . . , 𝜆𝑚)|2

∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚
∑︁

=

≤

(cid:12)
(cid:12)
(cid:12)

E (cid:2)|𝑋𝑚−1 ∗ 𝜓𝜆𝑚 |2(cid:3) 1/2 − E (cid:2)|𝑌𝑚−1 ∗ 𝜓𝜆𝑚 |2(cid:3) 1/2(cid:12)
2
(cid:12)
(cid:12)

E (cid:2)|(𝑋𝑚−1 − 𝑌𝑚−1) ∗ 𝜓𝜆𝑚 |2(cid:3)

E (cid:2)|𝑋𝑚−1 − 𝑌𝑚−1|2(cid:3)

E (cid:2)|(𝑋𝑚−2 − 𝑌𝑚−2) ∗ 𝜓𝜆𝑚−1 |2(cid:3)

E (cid:2)|𝑋𝑚−2 − 𝑌𝑚−2|2(cid:3)

(𝜆1,...,𝜆𝑚)∈Λ𝑚
∑︁

≤ 𝐶𝜓

(𝜆1,...,𝜆𝑚−1)∈Λ𝑚−1
∑︁

(𝜆1,...,𝜆𝑚−1)∈Λ𝑚−1
∑︁

(𝜆1,...,𝜆𝑚−2)∈Λ𝑚−2

≤ 𝐶𝜓

≤ 𝐶2
𝜓

...

≤ 𝐶𝑚

𝜓 E[|𝑋 − 𝑌 |2]

Now sum over all 𝑚 to get

∥𝑆2𝑋 − 𝑆2𝑋 ∥2

2 ≤

E[|𝑋 − 𝑌 |2] ≤ E[|𝑋 − 𝑌 |2].

∞
∑︁

𝐶𝑚

𝜓 E[|𝑋 − 𝑌 |2] =

𝐶𝜓
1 − 𝐶𝜓

𝑚=1
2 ≤ E[𝑋 2], which completes the proof.

Setting 𝑌 = 0 proves ∥𝑆2𝑋 ∥2

□

3.4.2 Diffeomorphism Contraction Estimates

Let 𝜏 be a stationary random process independent of 𝑋 such that ∥𝐷𝜏∥∞ ≤ 1

2𝑛 with probability
1. Define the deformed process 𝐿𝜏 𝑋 (𝑥) = 𝑋 (𝑥 − 𝜏(𝑥)), which is still stationary. We will need the

following lemma.

Lemma 35 ([11], Lemma 4.8). Let 𝐾𝜏 be an integral operator with a kernel 𝑘 𝜏 (𝑥, 𝑢) which depends

upon a random process 𝜏. If the following two conditions are satisfied:

and

E (cid:2)𝑘 𝜏 (𝑥, 𝑢)𝑘 ∗

𝜏 (𝑥, 𝑢′)(cid:3) = 𝑘 𝜏 (𝑥 − 𝑢, 𝑥 − 𝑢′)

∫

∫

R⋉

R⋉

|𝑘 𝜏 (𝑣, 𝑣′)||𝑣 − 𝑣′| 𝑑𝑣 𝑑𝑣′ < ∞,

64

then for any stationary process 𝑌 independent of 𝜏, E[|𝐾𝜏𝑌 (𝑥)|2] does not depend on 𝑥 and

E[|𝐾𝜏𝑌 |2] ≤ E[∥𝐾𝜏 ∥2]E[|𝑌 |2],

where ∥𝐾𝜏 ∥ is the operator norm in L2(R𝑛) for each realization of 𝜏.

Theorem 36 (Diffeomorphism Contraction Estimate). Consider the random process 𝑋 − 𝐿𝜏 𝑋. As-

sume that 𝛽𝐶𝜓 < 1/2 and ˆ𝑅𝑋−𝐿 𝜏 𝑋, the Fourier Transform of the covariance function, is bandlimited.

We have the following estimate for some 𝐶 > 0:

∥𝑆2𝑋 − 𝑆2𝐿𝜏 𝑋 ∥2

2 ≤ (𝐶 𝑀 2E[∥𝜏∥2

∞])E[|𝑋 |2].

Proof. Let 𝜙 be a function such that

ˆ𝜙(𝜔) =

1,

0,





𝜔 ∈ 𝐵1(0),

𝜔 ∉ 𝐵1(0).

Define 𝜙𝑀 (𝑥) = 𝑀 −𝑛𝜙(𝑀𝑥). Then we also know that ∫

R𝑛 𝜙𝑀 (𝑥) 𝑑𝑥 = 1.

Since our scattering operator is nonexpansive, we have

∥𝑆2𝑋 − 𝑆2𝐿𝜏 𝑋 ∥2

2 ≤ E[|𝑋 − 𝐿𝜏 𝑋 |2],

where the expectation is over all possible randomness. Notice that since ∫

R𝑛 𝜙𝑀 (𝑥) 𝑑𝑥 = 1, we can

write

E[|(𝑋 − 𝐿𝜏 𝑋) ∗ 𝜙𝑀 |2] =

=

=

∫

R𝑛

∫

R𝑛

∫

ˆ𝑅𝑋−𝐿 𝜏 𝑋 (𝜔)|𝜙𝑀 (𝜔)|2 𝑑𝜔 + E2 [(𝑋 − 𝐿𝜏 𝑋) ∗ 𝜙𝑀]

ˆ𝑅𝑋−𝐿 𝜏 𝑋 (𝜔)|𝜙𝑀 (𝜔)|2 𝑑𝜔 + E2 [𝑋 − 𝐿𝜏 𝑋]

ˆ𝑅𝑋−𝐿 𝜏 𝑋 (𝜔) 𝑑𝜔 + E2 [𝑋 − 𝐿𝜏 𝑋]

𝐵𝑀 (0)

= E[|𝑋 − 𝐿𝜏 𝑋 |2].

In other words, if we define 𝐴𝜙𝑅 𝑓 := 𝑓 ∗ 𝜙𝑅, we can write

E[|(𝑋 − 𝐿𝜏 𝑋) ∗ 𝜙𝑅 |2] = E[|( 𝐴𝜙𝑅 − 𝐴𝜙𝑅 𝐿𝜏) 𝑓 |2].

65

From estimates given in Theorem 3.6 of [23], in the deterministic case with 𝑓 ∈ L2(R𝑛) we have

∥( 𝐴𝜙𝑅 − 𝐴𝜙𝑅 𝐿𝜏) 𝑓 ∥2

2 ≤ 4𝑅2∥∇𝜙∥2

1∥𝜏∥2

∞∥ 𝑓 ∥2
2

,

where 𝜏 ∈ 𝐶1(R𝑛). It is proven in Appendix H of [11] that a operator of the form 𝐴𝜙𝑅 − 𝐴𝜙𝑅 𝐿𝜏 has

a kernel that satisfies Lemma 35. Thus, we have

E[|𝑋 − 𝐿𝜏 𝑋 |2] ≤ 4𝑅2∥∇𝜙∥2
1

E[∥𝜏∥2

∞]E[|𝑋 |2].

□

3.5 The Expected Scattering Transform When 1 < 𝑞 < 2

Now we generalize to the case where 𝑞 ∈ (1, 2). The case of 𝑞 = 1 has been addressed in [11].

The expected scattering transform is the set of all scattering moments:

𝑆𝑞 𝑋 = {𝑆𝑚

𝑞 𝑋 (𝜆1, . . . , 𝜆𝑚) : ∀(𝜆1, . . . , 𝜆𝑚) ∈ Λ𝑚, ∀𝑚 ∈ N}

(3.14)

with norm

∥𝑆𝑞 𝑋 ∥2

2 =

∞
∑︁

∑︁

𝑚=1

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚

𝑞 𝑋 (𝜆1, . . . , 𝜆𝑚)|2

and scattering distance

∥𝑆𝑞 𝑋 − 𝑆𝑞𝑌 ∥2

2 =

∞
∑︁

∑︁

𝑚=1

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚

𝑞 𝑋 (𝜆1, . . . , 𝜆𝑚) − 𝑆𝑚

𝑞 𝑌 (𝜆1, . . . , 𝜆𝑚)|2.

(3.15)

(3.16)

We start with a lemma that will help us determine when our generalized expected scattering

transform is well defined.

Lemma 37. Suppose 𝜓 is a littlewood paley wavelet. Then we have the following bound:

∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚

𝑞 𝑋 (𝜆1, . . . , 𝜆𝑚)|2 ≤ 𝛽𝑚𝐶𝑚

𝜓 𝜎2(𝑋) ≤ 𝛽𝑚𝐶𝑚

𝜓 E[𝑋 2].

66

Proof. Without a loss of generality, assume that 𝜓 is complex and remove 𝛽 from all the proofs.

For each 𝑚 ∈ N, we apply Jensen’s inequality to get

∑︁

|𝑆𝑚

𝑞 𝑋 (𝜆1, . . . , 𝜆𝑚)|2 =

∑︁

E (cid:2)||𝑋 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑚 |𝑞(cid:3) 2/𝑞

(𝜆1,...,𝜆𝑚)∈Λ𝑚

(𝜆1,...,𝜆𝑚)∈Λ𝑚
∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚

≤

E (cid:2)||𝑋 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑚 |2(cid:3)

= 𝛽𝑚𝐶𝑚

𝜓 𝜎2(𝑋)

≤ 𝛽𝑚𝐶𝑚

𝜓 E[𝑋 2].

□

Additionally, the expected scattering transform when 1 < 𝑞 < 2 are all nonexpansive operators

because of the following lemma.

Theorem 38. Suppose 𝜓 is a littlewood paley wavelet with 𝛽𝐶𝜓 ≤ 1

2. Then ∥𝑆𝑞 𝑋 − 𝑆𝑞𝑌 ∥2

2 ≤

E[|𝑋 − 𝑌 |2] and ∥𝑆𝑞 𝑋 ∥2

2 ≤ E[𝑋 2].

Proof. For notational simplicity, we use

𝑋𝑘 = ||𝑋 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑘 |,

𝑌𝑘 = ||𝑌 ∗ 𝜓𝜆1 | ∗ · · · | ∗ 𝜓𝜆𝑘 |.

We have

∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚

|𝑆𝑚

𝑞 𝑋 (𝜆1, . . . , 𝜆𝑚) − 𝑆𝑚

𝑞 𝑌 (𝜆1, . . . , 𝜆𝑚)|2

=

≤

≤

∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚
∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚
∑︁

(𝜆1,...,𝜆𝑚)∈Λ𝑚

E (cid:2)|𝑋𝑚−1 ∗ 𝜓𝜆𝑚 |𝑞(cid:3) 1/𝑞

(cid:12)
(cid:12)
(cid:12)

− E (cid:2)|𝑌𝑚−1 ∗ 𝜓𝜆𝑚 |𝑞(cid:3) 1/𝑞(cid:12)
(cid:12)
(cid:12)

2

E (cid:2)|(𝑋𝑚−1 − 𝑌𝑚−1) ∗ 𝜓𝜆𝑚 |𝑞(cid:3) 2/𝑞

E (cid:2)|(𝑋𝑚−1 − 𝑌𝑚−1) ∗ 𝜓𝜆𝑚 |2(cid:3)

≤ 𝐶𝑚

𝜓 E[|𝑋 − 𝑌 |2].

Now sum over all 𝑚 to finish the proof.

□

67

The following corollary also follows immediately from the proof above and the 𝑞 = 2 case.

Corollary 39. Suppose 𝜏 is a stochastic process independent of 𝑋 and 𝜓 is a littlewood paley

wavelet with 𝛽𝐶𝜓 ≤ 1/2. Consider the random process 𝑋 − 𝐿𝜏 𝑋, and suppose that the Fourier

Transform of its covariance function, ˆ𝑅𝑋−𝐿 𝜏 𝑋 (𝜔), is supported on some finite ball with radius 𝑅

centered at the origin: 𝐵𝑅 (0). We have the following estimate for some 𝐶 > 0:

∥𝑆𝑞 𝑋 − 𝑆𝑞 𝐿𝜏 𝑋 ∥2

2 ≤ 𝐶 𝑅2E[∥𝜏∥2

∞]E[|𝑋 |2].

68

CHAPTER 4

NONWINDOWED SCATTERING ON COMPACT RIEMANNIAN MANIFOLDS

In this chapter, we generalize our results with 𝑞 = 2 to compact Riemannian manifolds. First, let

us motivate why one would consider scattering transforms for non-Euclidean data. Suppose we

have number written on a set spheres (i.e. spherical MNIST). We would like to classify which

number is each of these spheres. A Euclidean approach would be to voxelize each of these spheres

as 𝑁 × 𝑁 × 𝑁 discretized cubes and feed these cubes into a feature extractor (i.e. a scattering

transform or a convolutional neural network). However, compared to a 𝑁 × 𝑁 image, this approach

is 𝑁 times more expensive in terms of memory because of the extra dimension. One can instead

consider these as signals on the sphere, which has a lower intrinsic dimension. The point is that

using Euclidean representations is not necessarily the best representation for feature extraction.

The paper [46] was the first to explore a unified framework for geometric deep learning,

and [28, 27, 29] provided a mathematical framework for scattering transforms for noneuclidean

data. Additionally, for spherical data, windowed scattering transforms have been generalized in

[47, 48], where the convolution operation is specific to the sphere, and numerical implementations

are optimized relative to [28] (with a trade-off of flexibility). As an aside, one could consider

nonwindowed versions of [47, 48] for classification tasks on the sphere.

In particular, [28] defines the nonwindowed scattering transform for compact manifolds as

L1 norms of a cascade of wavelet transforms and nonlinearities, which will be reviewed below.

Similar to scattering moments and nonwindowed scattering transforms for Euclidean data, one

would suspect that using L𝑞 norms instead of L1 norms provide richer discriptors for signals on

manifolds. This motivates our results for 𝑞 = 2. Other values of 𝑞 have been left to future work.

4.1 Notation for Scattering on Manifolds

Let M will be a compact, smooth, 𝑛-dimensional Riemannian manifold without boundary
contained in R𝑑, where 𝑑 ≥ 𝑛 with geodesic distance between two points 𝑥1, 𝑥2 ∈ M given by
𝑟 (𝑥1, 𝑥2) and Laplace-Beltrami operator denoted as Δ. The notation L𝑞 (M) denotes the set of
all functions 𝑓 : M → R such that ∫
| 𝑓 (𝑥)|𝑞 𝑑𝑥 < ∞, where 𝑑𝑥 is integration with respect to

M

69

the Riemannian volume. We use the notation Isom(M1, M2) be the set of isometries between

manifolds M1 and M2. Lastly, the set of diffeomorphisms on M will be denoted by Diff(M), and

the maximum placement of 𝛾 ∈ Diff(M) will be given by ∥𝛾∥∞ := sup𝑥∈M 𝑟 (𝑥, 𝛾(𝑥)).

4.2 Spectral Filters and the Geometric Wavelet Transform

We provide a brief summary of the geometric wavelet transform, as presented in [28]. The

convolution of 𝑓 , 𝑔 ∈ 𝐿2(R𝑛) is usually defined in space as

( 𝑓 ∗ 𝑔) (𝑥) =

∫

R𝑛

𝑓 (𝑦)𝑔(𝑥 − 𝑦) 𝑑𝑦.

However, for a general manifold, even under the conditions we have prescribed, a notation of

translation does not necessarily exist. Instead, one can consider a spectral definition of convolution

via the spectral decomposition of −Δ. Denote N ∪ {0} = N0. Because our manifold is compact, it

is well known that −Δ has a discrete spectrum, and we can order the eigenvalues in increasing order

and denote them as {𝜆𝑛}𝑛∈N0. We will denote the corresponding eigenfunctions as {𝜙𝑛 (𝑥)}𝑛∈N0,
which form an orthonormal basis for L2(M).

Suppose 𝑓 ∈ L2(M). Since the set of functions {𝜙𝑛 (𝑥)}𝑛∈N0 forms a basis in L2(M), we

decompose

𝑓 (𝑥) =

∑︁

⟨ 𝑓 , 𝜙𝑛⟩𝜙𝑛 (𝑥) =

(cid:18)∫

∑︁

𝑓 (𝑦)𝜙𝑛 (𝑦) 𝑑𝑦

(cid:19)

𝜙𝑛 (𝑥),

(4.1)

𝑛∈N0
which is similar to a Fourier series. Since 𝜙𝑛 (𝑦), is a replacement for a Fourier node, it is natural

𝑛∈N0

M

to let

ˆ𝑓 (𝑛) =

∫

M

𝑓 (𝑦)𝜙𝑛 (𝑦) 𝑑𝑦

and define convolution on M between functions 𝑓 , ℎ ∈ L2(M) as

𝑓 ∗ ℎ(𝑥) =

∑︁

𝑛∈N0

ˆ𝑓 (𝑛) ˆℎ(𝑛)𝜙𝑛 (𝑥).

(4.2)

(4.3)

Defining the operator 𝑇ℎ 𝑓 (𝑥) := 𝑓 ∗ ℎ(𝑥), it is easy to verify that the kernel for 𝑇ℎ is given by

𝐾ℎ (𝑥, 𝑦) :=

∑︁

𝑛∈N0

ˆℎ(𝑛)𝜙𝑛 (𝑥)𝜙𝑛 (𝑦).

(4.4)

70

Similar to how convolution commutes with translations on R𝑛, it is important for convolution

on M to be equivariant to a group action on M. The authors of [28] construct an operator by

convolving with functions that commute with isometries since the the geometry of M should be

preserved by a representation.

To accomplish this goal, we use a similar definition for spectral filters. A filter ℎ ∈ L2(M) is a

spectral filter if 𝜆𝑘 = 𝜆ℓ implies ˆℎ(𝑘) = ˆℎ(ℓ). One can prove that there exists 𝐻 : [0, ∞) → R such

that

𝐻 (𝜆𝑛) = ˆℎ(𝑛),

∀𝑛 ∈ N0.

Let 𝐺 : [0, ∞) → R be be nonnegative and decreasing with 𝐺 (0) > 0. A low-pass spectral filter

𝜙 is given in frequency as ˆ𝜙(𝑘) := 𝐺 (𝜆𝑘 ) and its dilation at scale 2 𝑗 for 𝑗 ∈ Z is ˆ𝜙 𝑗 (𝑘) := 𝐺 (2 𝑗𝜆𝑘 ).

Using the set of low pass filters, { ˆ𝜙 𝑗 } 𝑗 ∈Z, we define wavelets by

ˆ𝜓 𝑗 (𝑘) := (cid:2)| ˆ𝜙 𝑗−1(𝑘)|2 − | ˆ𝜙 𝑗 (𝑘)|2(cid:3) 1/2 ,

(4.5)

which is identical to standard constructions of Littlewood Paley wavelets in Euclidean Space.

Fix 𝐽 ∈ Z. Define the operators

𝐴𝐽 𝑓 := 𝑓 ∗ 𝜙𝐽,

Ψ𝑗 𝑓 := 𝑓 ∗ 𝜓 𝑗 ,

𝑗 ≤ 𝐽.

The windowed geometric wavelet transform is given by

𝑊𝐽 𝑓 := {𝐴𝐽 𝑓 , Ψ𝑗 𝑓 :

𝑗 ≤ 𝐽}

and the nonwindowed geometric scattering transform is given by

𝑊 𝑓 := {Ψ𝑗 𝑓 :

𝑗 ∈ Z}.

(4.6)

(4.7)

We have the following theorem, which provides a condition for when our wavelet frame is a

nonexpansive frame.

71

Theorem 40. Let 𝐺 :

[0, ∞) → R be nonnegative and decreasing with 0 < 𝐺 (0) = 𝐶,

lim𝑥→∞ 𝐺 (𝑥) = 0, and {𝜓 𝑗 } 𝑗 ∈Z is a set of wavelets generated by the low pass filter ˆ𝜙(𝑘) = 𝐺 (𝜆𝑘 ).

Then we have

∑︁

𝑗 ∈Z

∥ 𝑓 ∗ 𝜓 𝑗 ∥2

2 = 𝐶 ∥ 𝑓 ∥2
2

.

(4.8)

Proof. For fixed 𝐽 > 1, we telescope to get

𝐽
∑︁

𝑗=−𝐽

| ˆ𝜙 𝑗 (𝑘)|2 =

𝐽
∑︁

𝑗=−𝐽

(cid:2)|𝐺 (2 𝑗−1𝜆𝑘 )|2 − |𝐺 (2 𝑗𝜆𝑘 )|2(cid:3)

= |𝐺 (2𝐽−1𝜆𝑘 )|2 − |𝐺 (2−𝐽𝜆𝑘 )|2.

Since lim𝐽→∞ |𝐺 (2𝐽−1𝜆𝑘 )|2 and lim𝐽→∞ |𝐺 (2−𝐽𝜆𝑘 )|2 both exist, it follows that

∑︁

𝑗 ∈Z

| ˆ𝜙 𝑗 (𝑘)|2 = lim
𝐽→∞

|𝐺 (2𝐽−1𝜆𝑘 )|2 − lim
𝐽→∞

|𝐺 (2−𝐽𝜆𝑘 )|2 = 𝐶.

We can write

Thus, it follows that

∥ 𝑓 ∗ 𝜓 𝑗 ∥2

2 =

∑︁

𝑛∈N0

| ˆ𝜓 𝑗 (𝑘)|2| ˆ𝑓 (𝑘)|2.

∑︁

𝑗 ∈Z

∥ 𝑓 ∗ 𝜓 𝑗 ∥2

2 =

∑︁

∑︁

𝑗 ∈Z

𝑛∈N0

| ˆ𝑓 (𝑘)|2| ˆ𝜓 𝑗 (𝑘)|2

=

∑︁

𝑗 ∈Z

| ˆ𝑓 (𝑘)|2

(cid:32)

∑︁

𝑛∈N0

(cid:33)

| ˆ𝜓 𝑗 (𝑘)|2

= 𝐶 ∥ 𝑓 ∥2
2

.

□

4.3 The Geometric Scattering Transform

In an analogous manner to the Euclidean definition of the scattering transform, one would like

to find a representation that meaningfully encodes high frequency information of a signal 𝑓 . Define

the propagator as

𝑈 [ 𝑗] 𝑓 := |𝑊 𝑗 𝑓 |

∀ 𝑗 ∈ Z,

(4.9)

72

which is convolution of a wavelet and applying a nonlinearity. Similarly, we can define the

windowed propogator as

𝑈𝐽 [ 𝑗] 𝑓 := |𝑊 𝑗 𝑓 |

∀ 𝑗 ≤ 𝐽.

(4.10)

Similar to Scattering Transforms on Euclidean Space, one can apply a cascade of convolutions and

modulus operators repeatedly. In particular, for 𝑚 ∈ N, let 𝑗1, . . . , 𝑗𝑚 ∈ Z. The 𝑚-layer propogator

is defined as

𝑈 [ 𝑗1, . . . , 𝑗𝑚] := 𝑈 [ 𝑗𝑚] · · · 𝑈 [ 𝑗1] 𝑓 = | 𝑓 ∗ 𝜓 𝑗1 | ∗ 𝜓 𝑗2 · · · ∗ 𝜓 𝑗𝑚 |

(4.11)

and the 𝑚-layer windowed propogator is defined as

𝑈𝐽 [ 𝑗1, . . . , 𝑗𝑚] := 𝑈𝐽 [ 𝑗𝑚] · · · 𝑈𝐽 [ 𝑗1] 𝑓 := | 𝑓 ∗ 𝜓 𝑗1 | ∗ 𝜓 𝑗2 · · · ∗ 𝜓 𝑗𝑚 |,

𝑗1, . . . , 𝑗𝑚 ≤ 𝐽

(4.12)

with 𝑈 [∅] 𝑓 = 𝑓 and 𝑈𝐽 [∅] 𝑓 = 𝑓 . To aggregate low information and get local isometry invariance,

one can apply a low pass filter in a manner similar to pooling to each windowed propogator to get

windowed scattering coefficients:

𝑆 𝑗 [ 𝑗1, . . . , 𝑗𝑚] = 𝐴𝐽𝑈𝐽 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 = 𝑈𝐽 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 ∗ 𝜙𝐽,

where we defined 𝑆𝐽 [∅] 𝑓 = 𝑓 ∗ 𝜙𝐽. The windowed geometric scattering transform is given by

𝑆𝐽 𝑓 = {𝑆 𝑗 [ 𝑗1, . . . , 𝑗𝑚] : 𝑚 ≥ 0,

𝑗𝑖 ≤ 𝐽 ∀1 ≤ 𝑖 ≤ 𝑚}.

(4.13)

The authors of [28] were able to prove that this nonwindowed scattering operator was nonexpansive,

invariant to isometries up to the scale of the low pass filter, and stable to diffeomprohisms under

mild assumptions.

In addition, the authors consider a nonwindowed scattering transform, which removes the low

pass filtering. For applications such as manifold classification, requires full isometry invariance

instead of isometry invarance up to the scale 2𝐽. We see that

𝑆[ 𝑗1, . . . , 𝑗𝑚] 𝑓 (𝑥) = vol(M)−1/2∥𝑈 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 ∥1.

(4.14)

lim
𝐽→∞

As a proxy, it is more appropriate to consider

𝑆 𝑓 ( 𝑗1, . . . , 𝑗𝑚) = ∥𝑈 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 ∥1,

(4.15)

73

which motivates defining the nonwindowed geometric scattering transform as

𝑆 𝑓 = {𝑆[ 𝑗1, . . . , 𝑗𝑚] : 𝑚 ≥ 0,

𝑗𝑖 ∈ Z, ∀1 ≤ 𝑖 ≤ 𝑚}.

(4.16)

However, as mentioned previously, [36, 49] motivate the use of nonwindowed geometric scat-

tering operators as 2-norms of a cascade of convolutions and modulus operators:

𝑆𝑞 𝑓 ( 𝑗1, . . . , 𝑗𝑚) = ∥𝑈 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 ∥2.

Additionally, one can generalize nonwindowed geometric scattering transform to

𝑆2 𝑓 = {𝑆2 [ 𝑗1, . . . , 𝑗𝑚] : 𝑚 ≥ 0,

𝑗𝑖 ∈ Z, ∀1 ≤ 𝑖 ≤ 𝑚},

(4.17)

which we will call the 2-nonwindowed geometric scattering transform.

4.4 Generalizing Geometric Scattering Transforms

To measure stability and invariance properties of the 2-nonwindowed geometric scattering

transform, we need to define appropriate norms. The original nonwindowed geometric scattering

transform was a mapping ℓ2(L1(M)) → L2(M), but our interpretation is slightly different. In

particular, rather than thinking of the coefficients as a sequence, we group the coefficients in each

layer and define the norm

∥𝑆2 𝑓 ∥2 =

∞
∑︁

𝑚=1

(cid:169)
(cid:173)
(cid:171)

∑︁

( 𝑗1,..., 𝑗𝑚)∈Z𝑚

|𝑆2 𝑓 ( 𝑗1, . . . , 𝑗𝑚)|2(cid:170)
(cid:174)
(cid:172)

(4.18)

with scattering distance given by

∥𝑆2 𝑓 − 𝑆2𝑔∥2 =

∞
∑︁

𝑚=1

(cid:169)
(cid:173)
(cid:171)

∑︁

( 𝑗1,..., 𝑗𝑚)∈Z𝑚

|𝑆2 𝑓 ( 𝑗1, . . . , 𝑗𝑚) − 𝑆2𝑔( 𝑗1, . . . , 𝑗𝑚)|2(cid:170)
(cid:174)
(cid:172)

.

(4.19)

Theorem 41. Let 𝐺 : [0, ∞) → R be nonnegative and decreasing with 0 < 𝐺 (0) = 1√
2

,

lim𝑥→∞ 𝐺 (𝑥) = 0, and {𝜓 𝑗 } 𝑗 ∈Z be a set of spectral filters generated by 𝐺. Then we have

for all 𝑓 , 𝑔 ∈ L2(M).

∥𝑆2 𝑓 − 𝑆2𝑔∥ ≤ ∥ 𝑓 − 𝑔∥2

74

Proof. We begin by proving that

∑︁

( 𝑗1,..., 𝑗𝑚)∈Z𝑚

|𝑆2 𝑓 ( 𝑗1, . . . , 𝑗𝑚) − 𝑆2𝑔( 𝑗1, . . . , 𝑗𝑚)|2 ≤ 2−𝑚 ∥ 𝑓 ∥2
2

for all 𝑚 ∈ N via induction.

In the case of 𝑚 = 1, we see that

|𝑆2 𝑓 ( 𝑗) − 𝑆2𝑔( 𝑗)|2 =

∑︁

𝑗 ∈Z

≤

=

∑︁

𝑗 ∈Z
∑︁

𝑗 ∈Z
∑︁

𝑗 ∈Z

|∥ 𝑓 ∗ 𝜓 𝑗 ∥2 − |𝑔 ∗ 𝜓 𝑗 ∥2|2

∥ 𝑓 ∗ 𝜓 𝑗 − 𝑔 ∗ 𝜓 𝑗 ∥2
2

∥( 𝑓 − 𝑔) ∗ 𝜓 𝑗 ∥2
2

≤ 2−1∥ 𝑓 − 𝑔∥2
2

.

We can now work recursively. It follows that we can use similar ideas to the 𝑚 = 1 case to get

∑︁

( 𝑗1,..., 𝑗𝑚+1)∈Z𝑚+1

|𝑆2 𝑓 ( 𝑗1, . . . , 𝑗𝑚+1) − 𝑆2𝑔( 𝑗1, . . . , 𝑗𝑚+1)|2

∑︁

(cid:12)
(cid:12)∥𝑈 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 ∗ 𝜓 𝑗+1∥2 − ∥𝑈 [ 𝑗1, . . . , 𝑗𝑚]𝑔 ∗ 𝜓 𝑗+1∥2

(cid:12)
(cid:12)

2

( 𝑗1,..., 𝑗𝑚+1)∈Z𝑚+1
∑︁

(cid:12)
(cid:12)∥(𝑈 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 − 𝑈 [ 𝑗1, . . . , 𝑗𝑚]𝑔) ∗ 𝜓 𝑗+1∥2

(cid:12)
2
(cid:12)

=

=

( 𝑗1,..., 𝑗𝑚+1)∈Z𝑚+1

≤ 2−1 ∑︁

( 𝑗1,..., 𝑗𝑚)∈Z𝑚
∑︁

≤

( 𝑗1,..., 𝑗𝑚)∈Z𝑚
≤ 2−2 ∑︁

∥|𝑈 [ 𝑗1, . . . , 𝑗𝑚−1] 𝑓 ∗ 𝜓 𝑗𝑚 | − |𝑈 [ 𝑗1, . . . , 𝑗𝑚−1]𝑔 ∗ 𝜓 𝑗𝑚 |∥2
2

∥𝑈 [ 𝑗1, . . . , 𝑗𝑚−1] 𝑓 ∗ 𝜓 𝑗𝑚 − 𝑈 [ 𝑗1, . . . , 𝑗𝑚−1]𝑔 ∗ 𝜓 𝑗𝑚 ∥2
2

∥𝑈 [ 𝑗1, . . . , 𝑗𝑚−1] 𝑓 − 𝑈 [ 𝑗1, . . . , 𝑗𝑚−1]𝑔∥2
2

( 𝑗1,..., 𝑗𝑚−1)∈Z𝑚−1

≤ 2−𝑘+1∥ 𝑓 − 𝑔∥2
2

.

Now we can sum over all 𝑚 to get

∥𝑆2 𝑓 − 𝑆2𝑔∥2 ≤

∞
∑︁

𝑚=1

2−𝑚 ∥ 𝑓 − 𝑔∥2

2 = ∥ 𝑓 − 𝑔∥2
2

.

□

75

Corollary 42. Let 𝐺 : [0, ∞) → R be nonnegative and decreasing with 0 < 𝐺 (0) ≤ 1√
2

,

lim𝑥→∞ 𝐺 (𝑥) = 0, and {𝜓 𝑗 } 𝑗 ∈Z be a set of spectral filters generated by 𝐺. Then we have

for all 𝑓 ∈ L2(M).

∥𝑆2 𝑓 ∥ ≤ ∥ 𝑓 ∥2

Towards the point of embedding proper invariance, we provide a theorem that demonstrates that

the 2-nonwindowed geometric scattering transform is invariant to isometries.

Theorem 43. Let 𝜉 ∈ Isom(M, M′), and let 𝑓 ∈ L𝐿2(M). Define 𝑓 ′ = 𝑉𝜉 𝑓 and let 𝑆′

2 be the

corresponding 2-nonwindowed geometric scattering transform on M′ produced by a littlewood
paley wavelet satisfying the conditions described in Theorem 40. We have 𝑆′

𝑓 ′ = 𝑆2 𝑓 .

2

Proof. We see that 𝑆2 [∅] 𝑓 = ∥ 𝑓 ∥2 = ∥𝑉𝜉 𝑓 ∥2 since 𝑉𝜉 is an isometry. Now suppose that we

consider 𝑝 = ( 𝑗1, . . . , 𝑗𝑚). Then

𝑆2 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 = ∥𝑈 [ 𝑝] 𝑓 ∥2

= ∥𝑉𝜉𝑈 [ 𝑝] 𝑓 ∥2

= ∥𝑈 [ 𝑝]𝑉𝜉 𝑓 ∥2

= ∥𝑈 [ 𝑝] 𝑓 ′∥2
= 𝑆′

2 [ 𝑗1, . . . , 𝑗𝑚] 𝑓 ′.

Thus, we can see that 𝑆′

2

𝑓 ′ = 𝑆2 𝑓 .

□

Additionally, we also have a diffeomorphism stability result for 𝜆-bandlimited functions (i.e.

ˆ𝑓 )(𝑘) = ⟨ 𝑓 , 𝜙𝑘 ⟩ = 0 whenever 𝜆𝑘 > 𝜆).

Lemma 44 ([28]). Suppose 𝜉 ∈ Diff(M). If 𝑓 ∈ L2(M) is 𝜆-bandlimited, and 𝜉 ∈ Diff(M) can

be decomposed as 𝜉 = 𝜉1 ◦ 𝜉2, where 𝜉2 ∈ Diff(M) and 𝜉1 ∈ Isom(M), then

∥ 𝑓 − 𝑉𝜉 𝑓 ∥2 ≤ 𝐶 (M)𝜆𝑛 ∥𝜉 ∥∞∥ 𝑓 ∥2

for some constant 𝐶 (M).

76

Theorem 45. Let 𝑓 ∈ L2(M), and assume that 𝜓 is a wavelet family satisfying the conditions

of Theorem 41 with 𝐺 (𝜆) ≤ 𝑒−𝜆.

If 𝜉 ∈ Diff(M) can be decomposed as 𝜉 = 𝜉1 ◦ 𝜉2, where

𝜉2 ∈ Diff(M) and 𝜉1 ∈ Isom(M), then

∥𝑆2 𝑓 − 𝑆2𝑉𝜉 𝑓 ∥2 ≤ 𝐶 (M)𝜆𝑛 ∥𝜉 ∥∞∥ 𝑓 ∥2.

Proof. The transform is nonexpansive, so Lemma 44 gives the desired result.

□

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CHAPTER 5

CONCLUSIONS

This thesis has provided a generalization of nonwindowed scattering transforms to signals in

Euclidean space, as realizations stochastic processes, and signals on compact manifolds. Future

work involves the following:

• Generalize the diffeomorphism bound from chapter 2 to stochastic processes. This is possible,

but this is more difficult because the techniques used in Euclidean space for Chapter 2 do not

apply directly.

• Apply 𝑞-scattering moments to audio texture synthesis. Based on the results of [50], one

would expect that these scattering moments yield additional, relevant signal descriptors.

However, does this yield better signal synthesis?

• Generalize the results of chapter 2 to create nonwindowed scattering transforms as a cascade

of wavelet transforms, nonlinearities, and L𝑞 norms on a compact manifold. This is left to

future work, and requires results from singular integral theory.

78

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