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

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dc.contributor.author Sloane, Craig Andrew en_US
dc.date.accessioned 2011-09-22T17:48:16Z
dc.date.available 2011-09-22T17:48:16Z
dc.date.issued 2011-05-24 en_US
dc.identifier.uri http://hdl.handle.net/1853/41125
dc.description.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. en_US
dc.publisher Georgia Institute of Technology en_US
dc.subject Maz'ya en_US
dc.subject Hardy en_US
dc.subject Sobolev spaces en_US
dc.subject Inequalities en_US
dc.subject Rearrangments en_US
dc.subject Functional analysis en_US
dc.subject.lcsh Inequalities (Mathematics)
dc.subject.lcsh Fractional integrals
dc.title Hardy-Sobolev-Maz'ya inequalities for fractional integrals on halfspaces and convex domains en_US
dc.type Dissertation en_US
dc.description.degree Ph.D. en_US
dc.contributor.department Mathematics en_US
dc.description.advisor Committee Chair: Loss, Michael; Committee Member: Cvitanovic, Predrag; Committee Member: Geronimo, Jeff; Committee Member: Harrell, Evans; Committee Member: Iliev, Plamen en_US

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