On RR^n It is well known that the fourier transform is strong (p,p) for p in (1,2): There exists some $C>0$ such that for all $f\in L^p(\mathbb{R}^n)$ we have $$\Vert f \Vert_p \leq C \Vert \widehat{f} \Vert_p$$

-Where can I find a proof of this theorem over for nonabelian compact groups? Is it even true?

-Where can i find a proof of this theorem in the case of locally compact abelian groups? Is it even true?

Here is a special case: Let $\mathbb{A}$ be the group of Adeles, if $f\in L^p(\mathbb{A})$ is it true that $\widehat{f}\in L^p(\widehat{\mathbb{A}}).$