Hardy-Sobolev-Maz'ya inequalities for fractional integrals on halfspaces and convex domains

Abstract

This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation.

After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace.

Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.