The Hausdorff-Young inequality is one of the most fundamental results about the mapping properties of the Fourier transform: it says that
for all , where . It is important because it tells us that the Fourier transform maps continuously into , something which is not obvious when the exponent is not 1 or 2. When the underlying group is the torus, the corresponding Hausdorff-Young inequality is instead
The optimal constant is actually less than 1 in general, and it has been calculated for (and proven to be optimal) by Beckner, but this will not concern us here (if you want to find out what it is, take to be a gaussian). In the notes on Littlewood-Paley theory we also saw (in Exercise 7) that the inequality is false for 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.