Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?

Question

Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere. Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$. $\hat{f}$ is the Fourier transform fora function f.

Answer 1

If you interpret the $<\infty$ as "bounded by the $L^p$ norm of $f$" (as one normally does) then as quoted your statement is false. A correct version is found in

Tomas, Peter A., A restriction theorem for the Fourier transform, Bull. Am. Math. Soc. 81, 477-478 (1975). ZBL0298.42011.

States

Theorem Given a natural number $n$ and $p\in [1, \frac{2(n+1)}{n+3}]$, there exists a constant $c_{n,p}$ such that if $f \in L^p(\mathbb{R}^n)$ then $\int_{\mathbb{S}^{n-1}} |\hat{f}(\omega)|^2 ~d\omega < c_{n,p} \|f\|_{L^p}^2$.

This is usually called the Tomas-Stein restriction theorem, as Tomas proved it for $p < \frac{2(n+1)}{n+3}$ and Stein filled in the end point. Knapp's counterexamples show that the uniform bound is not possible when $p > \frac{2(n+1)}{n+3}$; these examples provide sequences of $L^p$ functions with norm $1$ for which the integral $\int_{\mathbb{S}^{n-1}} |\hat{f}(\omega)|^2 ~d\omega$ grow unboundedly.