Hausdorff-Young inequality and interpolation

The Hausdorff-Young inequality is one of the most fundamental results about the mapping properties of the Fourier transform: it says that

\displaystyle \| \widehat{f} \|_{L^{p'}(\mathbb{R}^d)} \leq \|f\|_{L^p(\mathbb{R}^d)}

for all {1 \leq p \leq 2}  , where \frac{1}{p} + \frac{1}{p'} = 1 . It is important because it tells us that the Fourier transform maps L^p continuously into L^{p'} , something which is not obvious when the exponent {p} is not 1 or 2. When the underlying group is the torus, the corresponding Hausdorff-Young inequality is instead

\displaystyle \| \widehat{f} \|_{\ell^{p'}(\mathbb{Z}^d)} \leq \|f\|_{L^p(\mathbb{T}^d)}.

The optimal constant is actually less than 1 in general, and it has been calculated for \mathbb{R}^d  (and proven to be optimal) by Beckner, but this will not concern us here (if you want to find out what it is, take {f} to be a gaussian). In the notes on Littlewood-Paley theory we also saw (in Exercise 7) that the inequality is false for {p} greater than 2, and we proved so using a randomisation trick enabled by Khintchine’s inequality1.

Today I would like to talk about how the Hausdorff-Young inequality (H-Y) is proven and how important (or not) interpolation theory is to this inequality. I won’t be saying anything new or important, and ultimately this detour into H-Y will take us nowhere; but I hope the ride will be enjoyable.

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Basic Littlewood-Paley theory II: square functions

This is the second part of the series on basic Littlewood-Paley theory, which has been extracted from some lecture notes I wrote for a masterclass. In this part we will prove the Littlewood-Paley inequalities, namely that for any {1 < p < \infty} it holds that

\displaystyle \|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 time there are also plenty more exercises, some of which I think are fairly interesting (one of them is a theorem of Rudin in disguise).
Part I: frequency projections.

4. Smooth square function

In this subsection we will consider a variant of the square function appearing at the right-hand side of (\dagger ) where we replace the frequency projections {\Delta_j} by better behaved ones.

Let {\psi} denote a smooth function with the properties that {\psi} is compactly supported in the intervals {[-4,-1/2] \cup [1/2, 4]} and is identically equal to {1} on the intervals {[-2,-1] \cup [1,2]}. We define the smooth frequency projections {\widetilde{\Delta}_j} by stipulating

\displaystyle  \widehat{\widetilde{\Delta}_j f}(\xi) := \psi(2^{-j} \xi) \widehat{f}(\xi);

notice that the function {\psi(2^{-j} \xi)} is supported in {[-2^{j+2},-2^{j-1}] \cup [2^{j-1}, 2^{j+2}]} and identically {1} in {[-2^{j+1},-2^{j}] \cup [2^{j}, 2^{j+1}]}. The reason why such projections are better behaved resides in the fact that the functions {\psi(2^{-j}\xi)} are now smooth, unlike the characteristic functions {\mathbf{1}_{[2^j,2^{j+1}]}}. Indeed, they are actually Schwartz functions and you can see by Fourier inversion formula that {\widetilde{\Delta}_j f = f \ast (2^{j} \widehat{\psi}(2^{j}\cdot))}; the convolution kernel {2^{j} \widehat{\psi}(2^{j}\cdot)} is uniformly in {L^1} and therefore the operator is trivially {L^p \rightarrow L^p} bounded for any {1 \leq p \leq \infty} by Young’s inequality, without having to resort to the boundedness of the Hilbert transform.
We will show that the following smooth analogue of (one half of) (\dagger ) is true (you can study the other half in Exercise 6).

Proposition 3 Let {\widetilde{S}} denote the square function

\displaystyle  \widetilde{S}f := \Big(\sum_{j \in \mathbb{Z}} \big|\widetilde{\Delta}_j f \big|^2\Big)^{1/2}.

Then for any {1 < p < \infty} we have that the inequality

\displaystyle  \big\|\widetilde{S}f\big\|_{L^p(\mathbb{R})} \lesssim_p \|f\|_{L^p(\mathbb{R})} \ \ \ \ \ (1)

holds for any {f \in L^p(\mathbb{R})}.

We will give two proofs of this fact, to illustrate different techniques. We remark that the boundedness will depend on the smoothness and the support properties of {\psi} only, and as such extends to a larger class of square functions.
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Fine structure of some classical affine-invariant inequalities and near-extremizers (account of a talk by Michael Christ)

Pdf version here: link.

I’m currently in Bonn, as mentioned in the previous post, participating to the Trimester Program organized by the Hausdorff Institute of Mathematics – although my time is almost over here. It has been a very pleasant experience: Bonn is lovely, the studio flat they got me is incredibly nice, Germany won the World Cup (nice game btw) and the talks were interesting. 2nd week has been pretty busy since there were all the main talks and some more unexpected talks in number theory which I attended. The week before that had been more relaxed instead, but I’ve followed a couple of talks then as well. Here I want to report about Christ’s talk on his work in the last few years, because I found it very interesting and because I had the opportunity to follow a second talk, which was more specific of the Hausdorff-Young inequality and helped me clarify some details I was confused about. If you get a chance, go to his talks, they’re really good.

What follows is an account of Christ’s talks – there are probably countless out there, but here’s another one. This is by no means original work, it’s very close to the talks themselves and I’m doing it only as a way to understand better. I’ll stick to Christ’s notation too. Also, I’m afraid the bibliography won’t be very complete, but I have included his papers, you can make your way to the other ones from there.

1. Four classical inequalities and their extremizers

Prof. Christ introduced four famous apparently unrelated inequalities. These are

  • the Hausdorff-Young inequality: for all functions {f \in L^p (\mathbb{R}^d)}, with {1\leq p \leq 2},

    \displaystyle \boxed{\|\widehat{f}\|_{L^{p'}}\leq \|f\|_{L^p};} \ \ \ \ \ \ \ \ \ \ \text{(H-Y)}

  • the Young inequality for convolution: if {1+\frac{1}{q_3}=\frac{1}{q_1}+\frac{1}{q_2}} then

    \displaystyle \|f \ast g\|_{L^{q_3}} \leq \|f\|_{L^{q_1}}\|g\|_{L^{q_2}};

    for convenience, he put it in trilinear form

    \displaystyle \boxed{ |\left\langle f\ast g, h \right\rangle|\leq \|f\|_{L^{p_1}}\|g\|_{L^{p_2}}\|h\|_{L^{p_3}}; } \ \ \ \ \ \ \ \ \ \ \text{(Y)}

    notice the exponents satisfy {\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=2} (indeed {q_1=p_1} and same for index 2, but {p_3 = q'_3});

  • the Brunn-Minkowski inequality: for any two measurable sets {A,B \subset \mathbb{R}^d} of finite measure it is

    \displaystyle \boxed{ |A+B|^{1/d} \geq |A|^{1/d} + |B|^{1/d}; } \ \ \ \ \ \ \ \ \ \ \text{(B-M)}

  • the Riesz-Sobolev inequality: this is a rearrangement inequality, of the form

    \displaystyle \boxed{ \left\langle \chi_A \ast \chi_B, \chi_C \right\rangle \leq\left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast} \right\rangle,} \ \ \ \ \ \ \ \ \ \ \text{(R-S)}

    where {A,B,C} are measurable sets and given set {E} the notation {E^\ast} stands for the symmetrized set given by ball {B(0, c_d |E|^{1/d})}, where {c_d} is a constant s.t. {|E|=|E^\ast|}: it’s a ball with the same volume as {E}.

These inequalities share a large group of symmetries, indeed they are all invariant w.r.t. the group of affine invertible transformations (which includes dilations and translations) – an uncommon feature. Moreover, for all of them the extremizers exist and have been characterized in the past. A natural question then arises

Is it true that if {f} (or {E}, or {\chi_E} where appropriate) is close to realizing the equality, then {f} must also be close (in an appropriate sense) to an extremizer of the inequality?

Another way to put it is to think of these questions as relative to the stability of the extremizers, and that’s why they are referred to as fine structure of the inequalities. If proving the inequality is the first level of understanding it, answering the above question is the second level. As an example, answering the above question for (H-Y) led to a sharpened inequality. Christ’s work was motivated by the fact that nobody seemed to have addressed the question before in the literature, despite being a very natural one to ask.

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