Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces

Abstract

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.