On the equivalence of fractional-order Sobolev semi-norms
Provides foundational theoretical results for fractional Sobolev spaces, benefiting researchers in PDEs and numerical analysis who rely on these norms.
The paper establishes equivalence relations between three types of fractional-order Sobolev semi-norms (Sobolev-Slobodeckij, interpolation, and quotient space) for orders between 0 and 1, showing they are uniformly equivalent under affine mappings that preserve shape regularity.
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.