Geometric characterization of generalized Hajłasz-Sobolev embedding domains

Abstract

In this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function ϕ:[0,∞)→[0,∞)\phi :\left[0,\infty )\to \left[0,\infty ). Given any bounded domain with the slice property, the authors prove that it is a generalized Hajłasz-Sobolev embedding domain if and only if it is a generalized FF-weak cigar domain, where FF is a modulus of continuity related to the weight function ϕ\phi of the generalized Hajłasz-Sobolev spaces under consideration. Comparing with the classical Hajłasz-Soblev spaces, one main difficulty in dealing with generalized Hajłasz-Sobolev spaces lies in that both its smoothness weight function ϕ\phi and the related modulus of continuity FF have no explicit expression. To overcome this, the authors introduce and use some key indices to accurately describe the increasing or the decreasing behavior of both ϕ\phi and FF. Besides the classical Hajłasz-Sobolev spaces, this result can be applied to many other nontrivial spaces such as Hajłasz-Sobolev spaces with logarithmic smoothness and is of wide generality.

Keywords

• hajłasz-sobolev embedding
• generalized smoothness
• cigar domain
• self-improving
• primary 46e35
• secondary 46e36
• 42b35
• 30l99