Sharp affine weighted L 2 Sobolev inequalities on the upper half space

Abstract

We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.