There is a p-adic analogue of Carleson's theorem. This was worked out by Hunt and Taibleson and you can find an exposition in Taibleson's book "Fourier analysis on local fields". (NB Taibleson did a lot of work extending Euclidean harmonic analysis results such as Carleson's theorem and boundedness of singular integrals to local fields.) I don't know if an adelic Carleson's theorem has been worked out.

The notion of smoothness in the p-adics is different than in real spaces since the p-adics are totally disconnected. Instead of say differentiability (which can be defined at least formally), one looks at locally constant functions. For more general groups, you need to have an appropriate notion(s) of smoothness to ask how changing the smoothness affects your problem.