Evolution equations and vector-valued Lp spaces: Strichartz estimates and symmetric diffusion semigroups. Taggart, Robert James en_US 2022-03-21T15:59:29Z 2022-03-21T15:59:29Z 2008 en_US
dc.description.abstract The results of this thesis are motivated by the investigation of abstract Cauchy problems. Our primary contribution is encapsulated in two new theorems. The first main theorem is a generalisation of a result of E. M. Stein. In particular, we show that every symmetric diffusion semigroup acting on a complex-valued Lebesgue space has a tensor product extension to a UMD-valued Lebesgue space that can be continued analytically to sectors of the complex plane. Moreover, this analytic continuation exhibits pointwise convergence almost everywhere. Both conclusions hold provided that the UMD space satisfies a geometric condition that is weak enough to include many classical spaces. The theorem is proved by showing that every symmetric diffusion semigroup is dominated by a positive symmetric diffusion semigoup. This allows us to obtain (a) the existence of the semigroup's tensor extension, (b) a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and (c) an holomorphic functional calculus for the extension's generator. The ergodic theorem is used to prove a vector-valued version of a maximal theorem by Stein, which, when combined with the functional calculus, proves the pointwise convergence theorem. The second part of the thesis proves the existence of abstract Strichartz estimates for any evolution family of operators that satisfies an abstract energy and dispersive estimate. Some of these Strichartz estimates were already announced, without proof, by M. Keel and T. Tao. Those estimates which are not included in their result are new, and are an abstract extension of inhomogeneous estimates recently obtained by D. Foschi. When applied to physical problems, our abstract estimates give new inhomogeneous Strichartz estimates for the wave equation, extend the range of inhomogeneous estimates obtained by M. Nakamura and T. Ozawa for a class of Klein--Gordon equations, and recover the inhomogeneous estimates for the Schrödinger equation obtained independently by Foschi and M. Vilela. These abstract estimates are applicable to a range of other problems, such as the Schrödinger equation with a certain class of potentials. en_US
dc.language English
dc.language.iso EN en_US
dc.publisher UNSW, Sydney en_US
dc.rights CC BY-NC-ND 3.0 en_US
dc.rights.uri en_US
dc.subject.other Strichartz estimates en_US
dc.subject.other Symmetric diffusion semigroups en_US
dc.subject.other Mathematical analysis en_US
dc.subject.other Evolution equations en_US
dc.subject.other Vector valued functions en_US
dc.title Evolution equations and vector-valued Lp spaces: Strichartz estimates and symmetric diffusion semigroups. en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Taggart, Robert James
dspace.entity.type Publication en_US
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Taggart, Robert James, Mathematics & Statistics, Faculty of Science, UNSW en_US School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
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