# Affine Invariant Surface Measure

In this short post I want to introduce an instance of certain objects that will be the subject of a few more posts. This particular object arises naturally in Affine Differential Geometry and turned out to have a relevant rôle in Harmonic Analysis too (in both Fourier restriction and in the theory of Radon transforms).

## 1. Affine Invariant measures

Affine Differential Geometry is the study of (differential-)geometric properties that are invariant with respect to $SL(\mathbb{R}^d)$. A very interesting object arising in Affine Geometry is the notion of an Affine Invariant Measure. Sticking to examples rather than theory (since the theory is still quite underdeveloped!), consider a hypersurface $\Sigma \subset \mathbb{R}^{d}$ sufficiently smooth to have well-defined Gaussian curvature, which we denote by $\kappa$ (a function on $\Sigma$). If we let $d\sigma$ denote the surface measure on $\Sigma$ (induced from the Lebesgue measure on the ambient space $\mathbb{R}^d$ for example, or by taking directly $d\sigma = d\mathcal{H}^{d-1}\big|_{\Sigma}$, the restriction of the $(d-1)$-dimensional Hausdorff measure to the hypersurface) then this crafty little object is called Affine Invariant Surface Measure and is given by

$\displaystyle d\Omega(\xi) = |\kappa(\xi)|^{1/(d+1)} \,d\sigma(\xi).$

It was first introduced by Blaschke for $d=3$ (finding the reference seems impossible; it’s [B] in this paper, if you feel luckier) and by Leichtweiss for general $d$. The reason this measure is so interesting is that it is (equi)affine invariant in the sense that if $\varphi(\xi) = A \xi + \eta$ is an equi-affine transformation (thus with $A \in SL(\mathbb{R}^d)$ and so volume-preserving since $\det A = \pm 1$) then, using subscripts to distinguish the two surfaces, we have

$\displaystyle \boxed{ \Omega_{\varphi(\Sigma)}(\varphi(E)) = \Omega_{\Sigma}(E) } \ \ \ \ \ \ \ (1)$

for any measurable $E \subseteq \Sigma$. We remark the following fact: that seemingly mysterious power $\frac{1}{d+1}$ in the definition of $d\Omega$ is the only exponent for which the resulting measure is (equi)affine-invariant.

# Basic Littlewood-Paley theory I: frequency projections

I have written some notes on Littlewood-Paley theory for a masterclass, which I thought I would share here as well. This is the first part, covering some motivation, the case of a single frequency projection and its vector-valued generalisation. References I have used in preparing these notes include Stein’s “Singular integrals and differentiability properties of functions“, Duoandikoetxea’s “Fourier Analysis“, Grafakos’ “Classical Fourier Analysis” and as usual some material by Tao, both from his blog and the notes for his courses. Prerequisites are some basic Fourier transform theory, Calderón-Zygmund theory of euclidean singular integrals and its vector-valued generalisation (to Hilbert spaces, we won’t need Banach spaces).

0. Introduction
Harmonic analysis makes a fundamental use of divide-et-impera approaches. A particularly fruitful one is the decomposition of a function in terms of the frequencies that compose it, which is prominently incarnated in the theory of the Fourier transform and Fourier series. In many applications however it is not necessary or even useful to resolve the function ${f}$ at the level of single frequencies and it suffices instead to consider how wildly different frequency components behave instead. One example of this is the (formal) decomposition of functions of ${\mathbb{R}}$ given by

$\displaystyle f = \sum_{j \in \mathbb{Z}} \Delta_j f,$

where ${\Delta_j f}$ denotes the operator

$\displaystyle \Delta_j f (x) := \int_{\{\xi \in \mathbb{R} : 2^j \leq |\xi| < 2^{j+1}\}} \widehat{f}(\xi) e^{2\pi i \xi \cdot x} d\xi,$

commonly referred to as a (dyadic) frequency projection. Thus ${\Delta_j f}$ represents the portion of ${f}$ with frequencies of magnitude ${\sim 2^j}$. The Fourier inversion formula can be used to justify the above decomposition if, for example, ${f \in L^2(\mathbb{R})}$. Heuristically, since any two ${\Delta_j f, \Delta_{k} f}$ oscillate at significantly different frequencies when ${|j-k|}$ is large, we would expect that for most ${x}$‘s the different contributions to the sum cancel out more or less randomly; a probabilistic argument typical of random walks (see Exercise 1) leads to the conjecture that ${|f|}$ should behave “most of the time” like ${\Big(\sum_{j \in \mathbb{Z}} |\Delta_j f|^2 \Big)^{1/2}}$ (the last expression is an example of a square function). While this is not true in a pointwise sense, we will see in these notes that the two are indeed interchangeable from the point of view of ${L^p}$-norms: more precisely, we will show that for any ${1 < p < \infty}$ it holds that

$\displaystyle \boxed{ \|f\|_{L^p (\mathbb{R})} \sim_p \Big\|\Big(\sum_{j \in \mathbb{Z}} |\Delta_j f|^2 \Big)^{1/2}\Big\|_{L^p (\mathbb{R})}. }\ \ \ \ \ (\dagger)$

This is a result historically due to Littlewood and Paley, which explains the name given to the related theory. It is easy to see that the ${p=2}$ case is obvious thanks to Plancherel’s theorem, to which the statement is essentially equivalent. Therefore one could interpret the above as a substitute for Plancherel’s theorem in generic ${L^p}$ spaces when ${p\neq 2}$.

In developing a framework that allows to prove ($\dagger$) we will encounter some variants of the square function above, including ones with smoother frequency projections that are useful in a variety of contexts. We will moreover show some applications of the above fact and its variants. One of these applications will be a proof of the boundedness of the spherical maximal function ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ (almost verbatim the one on Tao’s blog).

Notation: We will use ${A \lesssim B}$ to denote the estimate ${A \leq C B}$ where ${C>0}$ is some absolute constant, and ${A\sim B}$ to denote the fact that ${A \lesssim B \lesssim A}$. If the constant ${C}$ depends on a list of parameters ${L}$ we will write ${A \lesssim_L B}$.

# Some thoughts on the smoothing effect of convolution on measures

A question by Ben Krause, whom I met here at the Hausdorff Institute, made me think back of one of the earliest posts of this blog. The question is essentially how to make sense of the fact that the (perhaps iterated) convolution of a (singular) measure with itself is in general smoother than the measure you started with, in a variety of settings. It’s interesting to me because in this phase of my PhD experience I’m constantly trying to build up a good intuition and learn how to use heuristics effectively.

So, let’s take a measure ${\mu}$ on ${\mathbb{R}^d}$ with compact support (assume inside the unit ball wlog). We ask what we can say about ${\mu \ast \mu}$, or higher iterates ${\mu \ast \mu \ast \mu \ast \ldots}$ and more often1 about ${\mu \ast \tilde{\mu}}$. In particular, we’re interested in the case where ${\mu}$ is singular, i.e. its support has zero Lebesgue measure.

Before starting though, I would like to give a little motivation as to why such convolutions are interesting. Consider the model case where you have an operator defined by ${Tf = \sum_{j\in\mathbb{Z}}{T_j f}:= \sum_{j \in \mathbb{Z}}f\ast \mu_j}$ where ${\mu_j}$ are some singular measures and ${f \in L^2}$. One asks whether the operator ${T}$ is bounded on ${L^2}$, and the natural tool to use is Cotlar-Stein lemma, or almost-orthogonality (from which this blog takes its name). Then we need to verify that

$\displaystyle \sup_{j}{\sum_{k}{\|T_j T_k^\ast\|^{1/2}}} < \infty,$

and same for ${T^\ast_j T_k}$. But what is ${T_j T^\ast_k f}$? It’s simply

$\displaystyle T_j T^\ast_k f = f \ast d\tilde{\mu}_k \ast d\mu_j = f \ast (d\tilde{\mu}_k \ast d\mu_j),$

i.e. another convolution operator. Estimates on the convolution2 ${d\tilde{\mu}_k \ast d\mu_j}$ are likely to help estimate the norm of ${T_j T_k^\ast}$ then. But if this measure is not smooth enough, one can go forward, and since

$\displaystyle \|T T^\ast \| \leq \|T\|^{1/2} \|T^\ast T T^\ast\|^{1/2}$

one sees that estimates on ${d\tilde{\mu}_k \ast d\mu_j \ast d\tilde{\mu}_k }$ are likely to help, and so on, until a sufficient number of iterations gives a sufficiently smooth measure. This isn’t quite the iteration of a measure with itself, but in many cases one has an operator ${Tf = f \ast \mu}$ which then splits into the above sum by a spatial or frequency cutoff at dyadic scales. Then it becomes a matter of rescaling and the case ${d\tilde{\mu}_k \ast d\mu_j}$ can be reduced to that of ${d\tilde{\mu}_0 \ast d\mu_j}$ and further reduced to that of ${d\tilde{\mu}_0 \ast d\mu_0}$ by exploiting an iterate of the above norm inequality, namely that

$\displaystyle \|T_j T_0^\ast\|\leq \|T_j^\ast\|^{1-2^{\ell}}\|T_j (T_0^\ast T_0)^\ell\|^{2^{-\ell}}.$

Another possibility is to write ${d\nu = d\tilde{\mu}_k \ast d\mu_j}$ and consider working with ${d\nu \ast d\nu}$ instead, to obtain results in term of ${|j-k|}$. I will say more in the end, here I just wanted to show that they arise as natural objects.

# Weights theory basics, pt. II

This is the 2nd part of a post on basic weights theory. In the previous part I included definitions and fundamental facts, namely the nestedness of the ${A_p}$ classes, the weighted inequalities for ${M}$ and an extrapolation result for general operators. Refer to part I for notation and those results. This part includes instead a fundamental result on Calderón-Zygmund operators, together with inequalities relevant to the proof, and an elegant proof of Marcinkiewicz’s multiplier theorem as an application.

All of the following is taken from the excellent [Duo].

4. Calderón-Zygmund operators

This section is about the following result: if ${T}$ is now a Calderón-Zygmund operator, then ${T}$ is bounded on any ${L^p (A_p)}$ for all ${1; moreover, if ${p=1}$ then ${T}$ is weakly bounded on ${L^1 (A_1)}$. This is exactly what one would’ve hoped for, and suggests that the weights theory is indeed one worth pursuing.

Recall a CZ-operator is an ${L^2}$ bounded operator given by integration against a CZ-kernel ${K}$, defined on ${\mathcal{S}(\mathbb{R}^d)}$ at least, i.e.

$\displaystyle Tf(x) = \int{K(x,y)f(y)}\,dy,$

with the properties (for some ${\delta > 0}$)

1. ${|K(x,y)|\lesssim \frac{1}{|x-y|^n}}$;
2. ${|K(x,y)- K(x,y')|\lesssim \frac{|y-y'|^\delta}{(|x-y|+|x-y'|)^{d+\delta}}\qquad}$ for ${|y-y'| \leq \frac{1}{2}\max\{|x-y|,|x-y'|\}}$;
3. ${|K(x,y)- K(x',y)|\lesssim \frac{|x-x'|^\delta}{(|x-y|+|x'-y|)^{d+\delta}}\qquad}$ for ${|x-x'| \leq \frac{1}{2}\max\{|x-y|,|x'-y|\}}$.

Then, let’s state formally

Theorem 6 Let ${T}$ be a Calderón-Zygmund operator. Then ${T}$ is bounded on all ${L^p (A_p)}$ for ${1:

$\displaystyle \int{|Tf|^p w} \lesssim \int{|f|^p w}\qquad \forall w \in A_p.$

Moreover,

$\displaystyle w\left(\{|Tf|>\lambda\}\right) \lesssim \frac{1}{\lambda}\int{|f|w}\qquad \forall w \in A_1.$

# Weights theory basics, pt. I

Let’s work on ${\mathbb{R}^d}$ for some fixed dimension ${d}$. In the following I’ll write ${M}$ for the maximal function on uncentered cubes. In the following setting it won’t make a difference whether to work with this or with the usual one (defined on the centered balls), except in the constants; which I’ll throughout ignore thanks to my friend the wiggle notation and the fact that the dimension is fixed. In the proofs one has to resort alternatively to the dyadic, Hardy-Littlewood, centered versions of ${M}$ but it really just amounts to technicalities. I’ll ignore these minor issues as the reader can easily see how they are overcome. When in doubt, refer to the excellent [Duo], which is where all of the following is taken from.

I’ve split the post in two for convenience, you can find part II here.

1. Definition of ${A_p}$ and weighted inequalities for the maximal function ${M}$

Definition 1 (${A_p}$ weights) A locally integrable non negative function ${w}$ is said to be in the ${A_p}$ weight class if:

• in case ${p=1}$, it holds that for any ${x}$ and any cube ${Q \ni x}$

$\displaystyle \frac{w(Q)}{|Q|} \lesssim_w w(x) \ \ \ \ \ (1)$

(here ${w(Q)}$ is the measure of ${Q}$ w.r.t. measure ${w(y)\,dy}$, so ${w(Q)=\int_{Q}{w}\,dy}$);

• in case ${p>1}$, for any cube ${Q}$ it must be

$\displaystyle \left(\frac{1}{|Q|}\int_{Q}{w}\right)\left(\frac{1}{|Q|}\int_{Q}{w^{1-p'}}\right)^{p-1}\lesssim_w 1. \ \ \ \ \ (2)$

It is useful to refer to the quantity

$\displaystyle [w]_{A_p} :=\sup_{Q}{\left(\frac{1}{|Q|}\int_{Q}{w}\right)\left(\frac{1}{|Q|}\int_{Q}{w^{1-p'}}\right)^{p-1}}$

as the ${A_p}$ constant of ${w}$.

# Dimension of projected sets and Fourier restriction

I had a nice discussion with Tuomas after the very nice analysis seminar he gave for the harmonic analysis working group a while ago – he talked about the behaviour of Hausdorff dimension under projection operators and later we discussed the connection with Fourier restriction theory. Turns out there are points of contact but the results one gets are partial, and there are some a priori obstacles.

What follows is an account of the discussion. I will summarize his talk first.

1. Summary of the talk

1.1. Projections in ${\mathbb{R}^2}$

The problem of interest here is to determine whether there is any drop in the Hausdorff dimension of fractal sets when you project them on a lower dimensional vector space, and if so what can be said about the set of these “bad” projections. This is a very hard problem in general, so one has to start with low dimensions first. In ${\mathbb{R}^2}$ the projections are associated to the points in ${\mathbb{S}^1}$, namely for ${e\in\mathbb{S}^1}$ one has ${\pi_e (x) = (x\cdot e)e}$, and so for a given compact set ${K}$ of Hausdorff dimension ${0\leq \dim K \leq 1}$ one asks what can be said about the set of projections for which the dimension is smaller, i.e. ${\dim \pi(K) < \dim K}$. For ${s \leq \dim K}$, define the set of directions

$\displaystyle E_s (K):= \{e \in \mathbb{S}^1 \,:\, \dim \pi_e (K) < s\}.$

We refer to it as to the set of exceptional directions (of parameter ${s}$). One preliminary result is Marstrand’s theorem:

Theorem 1 (Marstrand) For any compact ${K}$ in ${\mathbb{R}^2}$ s.t. ${s<\dim K <1}$, one has

$\displaystyle |E_s (K)| = 0.$

In other words, the dimension is conserved for a.e. direction. The proof of the theorem relies on a characterization of dimension in terms of energy:

Theorem 2 (Frostman’s lemma) For ${K}$ compact in ${\mathbb{R}^d}$, it is ${s<\dim K}$ if and only if there exists a finite positive Borel measure ${\mu}$ supported in ${K}$ such that

$\displaystyle I_s(\mu):= \int_{K}{\int_{K}{\frac{d\mu(x)\,d\mu(y)}{|x-y|^s}}}<\infty.$

# Lorentz spaces basics & interpolation

(Updated with endpoint ${q = \infty}$)

I’ve written down an almost self contained exposition of basic properties of Lorentz spaces. I’ve found the sources on the subject to leave something to be desired, and I grew a bit confused at the beginning. Therefore this relatively short note (I might be ruining someone’s assignments out there, but I think the pros of writing down everything in one place balance the cons).

1. Lorentz spaces

In the following take ${1< p,q < \infty}$ otherwise specified, and ${(X, |\cdot|)}$ a ${\sigma}$-finite measure space with no atoms.

The usual definition of Lorentz space is as follows:

Definition 1 The space ${L^{p,q}(X)}$ is the space of measurable functions ${f}$ such that

$\displaystyle \|f\|_{L^{p,q}(X)}:= \left(\int_{0}^{\infty}{t^{q/p}{ f^\ast (t)}^q}\,\frac{dt}{t}\right)^{1/q} < \infty,$

where ${f^\ast}$ is the decreasing rearrangement [I] of ${f}$. If ${q=\infty}$ then define instead

$\displaystyle \|f\|_{L^{p,\infty}(X)} := \sup_{t}{t^{1/p} f^\ast (t)} < \infty.$