I'm using Muscalu and Schlag's book to study harmonic analysis by myself and I encountered the following problem:
Take an arbitrary function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ denotes the Schwartz space of functions. Take $\hat{f}$ to be the Fourier transform of $f$. Suppose that $\text{supp}(\hat{f}) \subset E \subset \mathbb{R}^d$ for some measurable set $E$. Prove that for any $1 \leq p \leq q \leq \infty$, the following estimate holds: $$||f||_{L^q} \leq |E|^{\frac{1}{p}-\frac{1}{q}}||f||_{L^p}$$ (Above $|E|$ denotes the Lebesgue measure of $E$)
The authors give the following hint: try proving the special case ($q=+\infty$ and $p=2$) at first and using duality and interpolation theorems.
The special case can be done by using Young's inequality and Plancherel identity. However, how can we apply duality of $L^p$ spaces here? Thanks for any suggestion in advance!