Sobolev Embedding Theorems and Their Generalizations for Maps Defined on Topological Spaces with Measures

For mappings of a topological space (X,μ) to a Banach space (Y,|·|Y) , we define analogs of Sobolev classes Wₚʳ(X;Y) , r=1,2,, as well as Sobolev–Slobodetsky classes Wₚʳ , r∈[1,∞) , and their generalizations. We prove theorems on the embedding into the scale of Lebesgue spaces Lq and into Orlizc spaces corresponding to rapidly increasing generating functions; other properties of Sobolev functions are studied as well.