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
And the above result can be generalized in special cases. For example, If
$R/Z$, the Carleson-Hunt theorem says the inversion formula holds a.e. for
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?