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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Kovač’s solution of the maximal Fourier restriction problem

About 2 years ago, Müller Ricci and Wright published a paper that opened a new line of investigation in the field of Fourier restriction: that is, the study of the pointwise meaning of the Fourier restriction operators. Here is an account of a recent contribution to this problem that largely sorts it out.

1. Maximal Fourier Restriction
Recall that, given a smooth submanifold \Sigma of \mathbb{R}^d with surface measure d\sigma , the restriction operator {R} is defined (initially) for Schwartz functions as

\displaystyle f \mapsto Rf:= \widehat{f}\Big|_{\Sigma};

it is only after having proven an a-priori estimate such as \|Rf\|_{L^q(\Sigma,d\sigma)} \lesssim \|f\|_{L^p(\mathbb{R}^d)} that we can extend {R} to an operator over the whole of L^p(\mathbb{R}^d), by density of the Schwartz functions. However, it is no longer clear what the relationship is between this new operator that has been operator-theoretically extended and the original operator that had a clear pointwise definition. In particular, a non-trivial question to ask is whether for d\sigma  -a.e. point \xi \in \Sigma we have

\displaystyle \lim_{r \to 0} \frac{1}{|B(0,r)|} \int_{\eta \in B(0,r)} |\widehat{f}(\xi - \eta)| d\eta = \widehat{f}(\xi), \ \ \ \ \ (1)

where B(0,r) is the ball of radius {r} and center {0} . Observe that the Lebesgue differentiation theorem already tells us that for a.e. element of \mathbb{R}^d in the Lebesgue sense the above holds; but the submanifold \Sigma has Lebesgue measure zero, and therefore the differentiation theorem cannot give us any information. In this sense, the question above is about the structure of the set of the Lebesgue points of \widehat{f} and can be reformulated as:

Q: can the complement of the set of Lebesgue points of \widehat{f} contain a copy of the manifold \Sigma ?

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Ptolemaics meeting #1

Together with some other PG students in the Harmonic Analysis working group, we’ve decided (it was Kevin’s idea originally) to set up a weekly meeting to learn about topics of harmonic analysis we don’t get to see otherwise (it works quite well as an excuse to drink beer, too). The topic we settled on arose pretty much by itself: it turned out that basically everybody was interested in time-frequency analysis on his own, either through Carleson’s theorem or some other related stuff. So we decided to learn about time-frequency analysis.

Last tuesday we had our first meeting: it was mainly aimed at discussing the arrangements to be made and what to read before next meeting, but we sketched some motivational introduction (it was quite improvised, I’m afraid); see below. Also, it was Odysseas that came up with the name. I think it’s quite brilliant: Ptolemy was the first to introduce the systematic use of epicycles in astronomy, and – as the science historian Giovanni Schiapparelli noticed – epicycles were nothing but the first historical appearance of Fourier series. That’s why they offered such accurate predictions even though the theory was wrong: by adding a suitable number of terms you can describe orbits within any amount of precision. Thus, from Carleson’s result you can go all the way back to Ptolemy: therefore Ptolemaics. Odysseas further added that Ptolemy’s first name was Claudius, like the roman emperor that first began the effective conquest of Britain; but that’s another story.

I will incorporate below a post I was writing for this blog about convergence of Fourier series, so it will be quite long in the end. Sorry about that, next posts will probably be way shorter.

1. Fourier series trivia

First some trivia of Fourier series as to brush up.

One wishes to consider approximations of functions (periodic of period 1) by means of trygonometric polynomials

\displaystyle \sum_{n=0}^{N}{\left(a_n \cos{2\pi n x} + b_n \sin{2\pi n x}\right)},

or, with a better notation,

\displaystyle \sum_{n=-N}^{N}{c_ n e^{2\pi i n x}}.

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