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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