  The primary aim of the paper is the study of Sobolev spaces in the context of
Gelfand pairs. The article commences with providing a historical overview and
motivation for the researched subject together with a summary of the current
state of the literature. What follows is a general outline of harmonic analysis
on Gelfand pairs, starting with a concept of positive-semidefinite functions,
through spherical functions and spherical transform and concluding with the
Hausdorff-Young inequality. The main part of the paper introduces the notion of
Sobolev spaces on Gelfand pairs and studies the properties of these spaces. It
turns out that Sobolev embedding theorems and Rellich-Kondrachov theorem still
hold true in this generalized context (if certain technical caveats are taken
into consideration).
