Probabilistic Proof of a Hausdorff-Young Type Inequality

Question

Let $1 \leq p <2$ and let $q$ be the Holder conjugate of $p$ so that $\frac{1}{p} + \frac{1}{q} = 1$. Show that for any $\epsilon >0$, there exists a Schwartz function $f \in S(\mathbb{R}^d)$, such that: $$ \|\hat{f}\|_{L^{q}(\mathbb{R}^d)} \leq \epsilon \|f\|_{L^p(\mathbb{R}^d)} $$ The exercise suggests that as a hint, one should use Khintchine's inequality: If $\epsilon_{n}$ is a IID sequence of $\mathrm{Unif}(\{-1,1\})$ random variables (random choice of signs) and $x_n$ is (finite) sequence of complex numbers we have a constant $C(p) >0$ with: $$ \frac{1}{C(p)}\left(\sum_{n = 1}^{N}|x_n|^2\right)^{1/2} \leq \left(\mathbb{E}\left[\left(\sum_{n = 1}^N\epsilon_nx_n\right)^p\right]\right)^{1/p} \leq C(p)\left(\sum_{n = 1}^{N}|x_n|^2\right)^{1/2} $$ Does anyone have any ideas as to how one should properly apply this inequality?

Answer 1

We exploit a trick called randomisation. Broadly, the idea of the trick is to introduce some random signs into a sum and then use the Khintchine inequality to see that there is some deterministic choice of those signs that has some desired behaviour.

Fix a non-negative, non-zero smooth function $\varphi$ supported in the unit ball.

Now fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ supporting a sequence of IID Rademacher random variables $(\epsilon_n)_{n \geq 1}$ as suggested in the hint. We also fix for now an $N$ which will later be taken to be suitably large (in a way that depends only on $\varepsilon$, $p$ and $\varphi$).

Choose points $x_1, \dots, x_N$ such that $\varphi_j(\cdot) = \varphi(\cdot - x_j)$ have disjoint support. Define for $\omega \in \Omega$, $$\Phi_\omega(x) = \sum_{j=1}^N \epsilon_j(\omega) \varphi_j(x)$$

First note that by the disjoint support condition we have that $\|\Phi_\omega\|_{L^p} \sim N^{1/p}$ where the constant depends only on the choice of $\varphi$.

Also $$\mathbb{E} \left[ \|\hat{\Phi}_\omega\|_{L^q}^q \right ] = \mathbb{E}\left[ \int \left | \sum_{j=1}^N \epsilon_j e^{-2\pi i \langle x_j, x \rangle}\hat{\varphi}(x) \right|^q dx \right] \lesssim \int \left(\sum_{j=1}^N |\hat{\varphi}(x)|^2 \right)^{q/2} dx \sim N^{q/2}$$ where the inequality follows by applying Fubini's theorem followed by Khintchine's inequality. It follows that there exists a fixed $\omega \in \Omega$ (which just means a choice of signs for the $\epsilon_j$) such that $$\|\hat{\Phi}_\omega\|_{L^q} \lesssim N^{1/2}.$$

Hence for this $\omega$, $$\frac{\|\hat{\Phi}_\omega\|_{L^q}}{\|\Phi_\omega\|_{L^p}} \lesssim N^{\frac12 - \frac1p}.$$

Since $1 \leq p <2$, the right hand side of this goes to $0$ as $N \to \infty$. This means that the argument shows that for every $N$ there is a $\Phi_\omega$ such that inequality holds with an implicit constant that is independent of $N$, which means that by taking sufficiently large $N$ we get the desired result.