Hardy-Sobolev-Maz'ya Inequalities for Fractional Integrals on Halfspaces and Convex Domains

Dissertation Defense
Tuesday, April 19, 2011 - 4:00pm for 1 hour (actually 50 minutes)
Skiles 006
Craig A. Sloane – School of Mathematics, Georgia Tech
Michael Loss
Classical Hardy, Sobolev, and Hardy-Sobolev-Maz'ya inequalities are well known results that have been studied for awhile. In recent years, these results have been been generalized to fractional integrals. This Dissertation proves a new Hardy inequality on general domains, an improved Hardy inequality on bounded convex domains, and that the sharp constant for any convex domain is the same as that known for the upper halfspace. We also prove, using a new type of rearrangement on the upper halfspace, based in part on Carlen and Loss' concept of competing symmetries, the existence of the fractional Hardy-Sobolev-Maz'ya inequality in the case p = 2, as well as proving the existence of minimizers, at least in limited cases.