Some thoughts on the smoothing effect of convolution on measures

Pdf version here: link.

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.

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Weights theory basics, pt. II

Pdf version here: link.

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<p<\infty}; 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<p<\infty}:

\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.

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Ptolemaics meetings 4 & 5 & 6 ; pt I

These last ones have been quite interesting meetings, I’m happy about how the whole thing is turning out. Sadly I’m very slow at typing and working out the ideas, so I have to include three different meetings in one. Since the notes are getting incredibly long, I’ll have to split it in at least two parts.I include the pdf version of it, in case it makes it any easier to read.

ptolemaics meeting 4 & 5 & 6 pt I

Let me get finally into the time frequency of the Walsh phase plane. I won’t include many proofs as they are already well written in Hytönen’s notes (see previous post). My main interest here is the heuristic interpretation of them (disclaimer: you might think I’m bullshitting you at a certain point, but I’m probably not). Ideally, it would be very good to be able to track back the train of thoughts that went in Fefferman’s and Thiele-Lacey’s proofs.

Sorry if the pictures are shit, I haven’t learned how to draw them properly using latex yet.

1. Brush up

Recall we have Walsh series for functions {f \in L^2(0,1)} defined by

\displaystyle W_N f(x) = \sum_{n=0}^{N}{\left\langle f,w_n\right\rangle w_n(x)},

the (Walsh-)Carleson operator here is thus

\displaystyle \mathcal{C}f(x) = \sup_{N\in \mathbb{N}}{|W_N f(x)|},

and in order to prove {W_N f(x) \rightarrow f(x)} a.e. for {N\rightarrow +\infty} one can prove that

\displaystyle \|\mathcal{C}f\|_{L^{2,\infty}(0,1)} \lesssim \|f\|_{L^2(0,1)}.

There’s a general remark that should be done at this point: the last inequality is equivalent to

\displaystyle \left|\left\langle\mathcal{C}f, \chi_E\right\rangle\right| = \left|\int_{E}{\mathcal{C}f}\,dx\right| \lesssim |E|^{1/2}\|f\|_{L^2(0,1)}

to hold on every measurable {E} (of finite measure).
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