I am asking this question to try to decipher notes given by my professor.
We have a function, $f: \mathbb{R}^2 \rightarrow \mathbb{R} $, and we are trying to bound its $L^p$-Norm, where $p>2$. This function has a nice Fourier Transform, so we would like to bound the norm of $f$ by the norm of $\mathcal{F}[f] = \hat{f}$ somehow.
According to my professor's scrawlings, we can say that, for $p>2$,
$ ||f||_{p} \leq C ||\hat{f}||_{\frac{p}{p-1}} $, by "Riesz Interpolation".
I am unfortunately not able to figure what he means by this, or how this can be proven. He writes that it somehow comes from the fact that the $L^\infty$- and $L^2$-norms can be bounded in the following ways:
$||f||_{\infty} \leq C ||\hat{f}||_{1}, \ \ ||f||_{2} = ||\hat{f}||_{2}$.
I understand these two facts well enough, but am not seeing how we get the more general result above.
Your professor is referring to the Riesz-Thorin interpolation theorem. It basically says that if a linear operator $T$ is bounded as a map $L^{p_1}\to L^{q_1}$ and also as a map $L^{p_2}\to L^{q_2}$, then it is also bounded as a map $L^{p_\theta}\to L^{q_\theta}$ for $$ \frac{1}{p_\theta} = \frac{\theta}{p_1} + \frac{1-\theta}{p_2}, \frac{1}{q_\theta} = \frac{\theta}{q_1} + \frac{1-\theta}{q_2}, ~\theta\in[0,1]. $$ The full statement also gives you a bound on the operator norm. The idea is to apply this in the specific case that $T$ is the Fourier transform, using the two endpoint bounds you have already stated.
The resulting estimate is called the Hausdorff-Young inequality.