Higher-order Sobolev embeddings and isoperimetric inequalities

Abstract

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities.

Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rn, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces.

Keywords

• Isoperimetric function
• Higher-order Sobolev embeddings
• Rearrangement-invariant spaces
• John domains
• Maz’ya domains
• Product probability measures
• Gaussian Sobolev inequalities
• Hardy operator