On the equivalence of fractional-order Sobolev semi-norms

TL;DR

This paper investigates the relationships and equivalences among different fractional-order Sobolev semi-norms, establishing conditions under which they are uniformly equivalent under affine transformations, with implications for shape-regular domains.

Contribution

It provides new mutual estimates and mapping properties of fractional Sobolev semi-norms, clarifying their equivalence under affine transformations for shape-regular domains.

Findings

Mutual estimates of Sobolev-Slobodeckij, interpolation, and quotient space semi-norms.

Uniform equivalence of semi-norms under affine mappings for shape-regular domains.

Enhanced understanding of fractional Sobolev space mappings and equivalences.

Abstract

We present various results on the equivalence and mapping properties under affine transformations of fractional-order Sobolev norms and semi-norms of orders between zero and one. Main results are mutual estimates of the three semi-norms of Sobolev-Slobodeckij, interpolation and quotient space types. In particular, we show that the former two are uniformly equivalent under affine mappings that ensure shape regularity of the domains under consideration.