The sharp affine L2 Sobolev trace inequality and affine energy in the fractional Sobolev spaces

Abstract

We establish an affine invariant version of the sharp L2 Sobolev trace inequality which contains a recent result of De Nápoli et al. [9] as special case. We also give a notion of the so-called affine fractional energy for functions in the fractional homogeneous Sobolev space H˙s(Rn) for s∈(0,1) and n>2s. We investigate some properties of this affine fractional energy such as a representation formula and the affine fractional Pólya–Szegö principle. Finally, we obtain a sharp affine fractional L2 Sobolev inequality in the fractional homogeneous Sobolev space H˙s(Rn) for s∈(0,1) and n>2s.