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dc.contributor.advisorWright, Jim
dc.contributor.advisorGimperlein, Heiko
dc.contributor.authorWheeler, Reuben
dc.date.accessioned2022-03-22T11:16:08Z
dc.date.available2022-03-22T11:16:08Z
dc.date.issued2022-03-21
dc.identifier.urihttps://hdl.handle.net/1842/38783
dc.identifier.urihttps://doi.org/10.1007/s00041-020-09798-8en
dc.identifier.urihttp://dx.doi.org/10.7488/era/2037
dc.description.abstractFor a smooth surface Γ of arbitrary codimension, one can consider the Lp mapping properties of the Bochner-Riesz multiplier m(ζ) = dist(ζ,Γ)^α φ(ζ), where α > 0 and φ is an appropriate smooth cutoff function. Even for the sphere, the exact Lp boundedness range remains a central open problem in Euclidean harmonic analysis. We consider the Lp integrability of the Bochner-Riesz convolution kernel for a particular class of surfaces (of any codimension). For a subclass of these surfaces the range of Lp integrability of the kernels differs substantially from the Lp boundedness range of the corresponding Bochner-Riesz multiplier operator. Extending work of Mockenhoupt, we then establish a range of operator bounds, which are sharp in the α exponent, under the assumption of an appropriate L2 restriction estimate. Hickman and Wright established sharp oscillatory integral estimates, associated with a particular class of surfaces, and derived restriction estimates. We extend this work to certain curves of standard type and corresponding surfaces of revolution. These surfaces are discussed as an explicit class for which we have Lp → Lp boundedness of the corresponding Bochner-Riesz operators. Understanding the structure of the roots of real polynomials is important in obtaining stable bounds for oscillatory integrals with polynomial phases. For real polynomials with exponents in some fixed set, Ψ(t)=x+y1 t^{k1} +...+yL t^{kL}, we analyse the different possible root structures that can occur as the coefficients vary. We first establish a stratification of roots into tiers containing roots of comparable sizes. We then show that at most L non-zero roots can cluster about a point. Supposing additional restrictions on the coefficients, we derive structural refinements. These structural results extend work of Kowalski and Wright and provide a characteristic picture of root structure at coarse scales. As an application, these results are used to recover the sharp oscillatory integral estimates of Hickman and Wright, using bounds for oscillatory integrals of Phong and Stein.en
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en
dc.language.isoenen
dc.publisherThe University of Edinburghen
dc.relation.hasversionR. Wheeler, A note on the L p integrability of a class Bochner-Riesz kernels, J. Fourier An. Appl. 26 (2020), no. 86.en
dc.subjectBochner-Riesz conjectureen
dc.subjectoscillatory integralsen
dc.subjectLp integrabilityen
dc.subjectFourier restrictionen
dc.subjectpolynomial rootsen
dc.subjectroot findingen
dc.titleMethods for the analysis of oscillatory integrals and Bochner-Riesz operatorsen
dc.typeThesis or Dissertationen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen


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