On the equivalence of fractional-order Sobolev semi-norms

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.