Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted L spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric into Lq with respect to another measure. The considered can be fractional order some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences covering families local Poincaré inequalities. show how construct such suitable coverings For locally doubling measures, we prove self-improvement property for two-weighted inequalities, which applies lower-dimensional simultaneously treat various spaces, as the Newtonian, Hajłasz rather measures sets, including fractals domains fractal boundaries. By considering boundaries domains, obtain trace above spaces. In case Newtonian exactly characterize when measure compact. tools illustrated by concrete examples. satisfying dimension conditions, recover several classical embedding theorems sets Rn.