What is the most general manifold for which a Schwartz space like object is defined and that space can be given a structure analogous to the convolution algebra structure it is typically equipped in vector spaces (over $\mathbb{R}$ and $\mathbb{C}$)? Are there more examples than uni-modular Lie groups where the convolution theorem holds?

Compact Surfaces in $\mathbb{R}^3$ are cases of most interest.

Can one construct for all/any compact Surfaces in $\mathbb{R}^3$ a Schwartz space like object in a way so that the space can be given a structure analogous to the convolution algebra structure it is typically equipped in vector spaces (over $\mathbb{R}$ and $\mathbb{C}$)?

Does anything analogous to the convolution theorem hold in such a setting?