Exercise 8, § 1

Let $p\in [1,2]$ and $q\in [2,+\infty ]$ be such that $1/p+ 1/q= 1$.

a) Let $f\in \mathscr{K}(G)$. Now prove that $\|\mathscr{F}(f)\|_q\leqslant \|f\|_p$. (This inequality is valid for $p= 1$ and $p= 2$. In the general case, use the inequality of M. Riesz (INT, IV, §6, exerc. 18).)

b) Deduce that the restriction of $\mathscr{F}$ to $\mathscr{K}(G)$ extends to a continuous linear mapping from $L^p(G)$ into $L^q(\widehat{G})$ which coincides with the Fourier transform on $L^p(G)\cap L^1(G)$ and on $L^p(G)\cap L^2(G)$.