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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F -weak cigar domain, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F is a modulus of continuity related to the weight function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> \phi and the related modulus of continuity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> \phi and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F . 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.