As we know, the inversion formula of Fourier transformation holds pointwise for Schwartz class. We also have a general result concerning the inversion of Fourier transformation on locally compact abelian groups, which says that if $f$ belongs to the intersection of the $L^1$-algebra and the Fourier-Stieltjes algebra on a locally compact abelian group $G$, then the inversion formula holds a.e. for $f$. And the above result can be generalized in special cases. For example, If $G$ is $R$ or $R/Z$, the Carleson-Hunt theorem says the inversion formula holds a.e. for $f$ in $L^p$ with $1<p<\infty$.

My question is, is there any other version of generalization of inversion of Fourier transformation concerning a given locally compact abelian group $G$? For example, $G$ is an abelian Lie group, or $G$ is a compact group?

abeliangroups -- Fourier analysis on nonabelian compact Lie groups has been much studied but opens up a whole new can of worms $\endgroup$