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}.

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Dimension of projected sets and Fourier restriction

Pdf version here: link.

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.

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