# Probabilistic Proof of a Hausdorff-Young Type Inequality

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?

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.