Interlude: Atomic decomposition of L(log L)^r

This is going to be a shorter post about a technical fact that will be used in concluding the proof of the Tao-Wright lemma.
What we are going to see today is an atomic decomposition of the Orlicz spaces of L (\log L)^r type. Surprisingly, I could find no classical references that explicitely state this useful little fact – some attribute it to Titchmarsh, Zygmund and Yano; indeed, something resembling the decomposition can be found for example in Zygmund’s book (Volume II, page 120). However, I could only find a proper statement together with a proof in a paper of Tao titled “A Converse Extrapolation Theorem for Translation-Invariant Operators“, where he claims it is a well-known fact and proves it in an appendix (the paper is about reversing the implication in an old extrapolation theorem of Yano [1951], a theorem that tells you that if the operator norms \|T\|_{L^p \to L^p} blow up only to finite order as p \to 1^{+} , then you can “extrapolate” this into an endpoint inequality of the type \|Tf\|_{L^1} \lesssim \|f\|_{L(\log L)^r} ).

Briefly stated, the result is as follows. We will consider only L(\log L)^r ([0,1]), that is the Orlicz space of functions on [0,1] with Orlicz/Luxemburg norm

\displaystyle \|f\|_{L(\log L)^r ([0,1])} = \inf \Big\{\mu > 0 \text{ s.t. } \int_{0}^{1} \frac{|f(x)|}{\mu} \Big(\log \Big(2 + \frac{|f(x)|}{\mu}\Big)\Big)^{r} \,dx \leq 1 \Big\}.

Our atoms will be quite simply normalised characteristic functions: that is, for any measurable set E \subset [0,1] we let a_E denote the atom associated to E , given by

\displaystyle a_E := \frac{\mathbf{1}_E}{\|\mathbf{1}_E\|_{L(\log L)^r}};

obviously \|a_E\|_{L(\log L)^r} = 1 .
The statement is then the following.

Atomic decomposition of L(\log L)^r :
Let f \in L(\log L)^{r}([0,1]) . Then there exist measurable sets (E_j)_j and coefficients (\alpha_j)_j such that

\displaystyle f = \sum_{j} \alpha_j a_{E_j}

and

\displaystyle \sum_{j} |\alpha_j| \lesssim \|f\|_{L(\log L)^r}.

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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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Christ’s result on near-equality in Riesz-Sobolev inequality

Pdf: link.

It’s finally time to address one of Christ’s papers I talked about in the previous two blogposts. As mentioned there, I’ve chosen to read the one about the near-equality in the Riesz-Sobolev inequality because it seems the more approachable, while still containing one very interesting idea: exploiting the additive structure lurking behind the inequality via Freiman’s theorem.

1. Elaborate an attack strategy

Everything is in dimension {d=1} and some details of the proof are specific to this dimension and don’t extend to higher dimensions. I’ll stick to Christ’s notation.

Recall that the Riesz-Sobolev inequality is

\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} \ \ \ \ \ (1)

and its extremizers – which exist under the hypothesis that the sizes are all comparable – are intervals, i.e. the intervals are the only sets that realize equality in (1). See previous post for further details. The aim of paper [ChRS] is to prove that whenever {\left\langle \chi_{A} \ast \chi_{B}, \chi_{C}\right\rangle} is suitably close to {\left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast}\right\rangle} (i.e. we nearly have equality) then the sets {A,B,C} are nearly intervals themselves.

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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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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).

Here’s a link to the pdf version of this post: Lorentz spaces primer

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.

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