I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim:

Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ denotes the Schwartz space of functions. Let $\hat{f}$ denote the Fourier transform of $f$. Assume that there exists some measurable set $E$, such that $\text{supp}(\hat{f}) \subset E \subset \mathbb{R}^d$. Then for any $1 \leq p \leq q \leq \infty$, we have the following inequality: ($|E|$ below denotes the Lebesgue measure of $E$) $$||f||_{L^q} \leq |E|^{\frac{1}{p}-\frac{1}{q}}||f||_{L^p}$$ I have managed to show the special case when $q=+\infty$ and $p=2$ by using Young's inequality and Plancherel identity. However, the hint says that we still need to use duality and interpolation to deduce the general conclusion. Any ideas on this?

Moreover, how might this estimate be related to the probability version of Bernstein inequality? Thanks in advance!