Topics in affine and discrete harmonic analysis
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Date
01/07/2015Author
Hickman, Jonathan Edward
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Abstract
In this thesis a number of problems in harmonic analysis of a geometric flavour are discussed
and, in particular, the Lebesgue space mapping properties of certain averaging and Fourier
restriction operators are studied. The first three chapters focus on the perspective afforded by
affine-geometrical considerations whilst the remaining chapter considers some discrete variants
of these problems.
In Chapter 1 there is an overview of the basic affine theory of the aforementioned operators
and, in particular, the affine arc-length and surface measures are introduced.
Chapter 2 presents work of the author, submitted for publication, concerning an operator
which takes averages of functions on Euclidean space over both translates and dilates of a fixed
polynomial curve. Moreover, the averages are taken with respect to the affine arc-length; this
allows one to prove Lebesgue space estimates with a substantial degree of uniformity in the
constants. The sharp range of uniform estimates is obtained in all dimensions except for an
endpoint.
Chapter 3 presents some work of the author, published in Mathematika, concerning a family
of Fourier restriction operators closely related to the averaging operators discussed in Chapter
2. Specifically, a Fourier restriction estimate is obtained for a broad class of conic surfaces by
introducing a certain measure which exhibits a special kind of affine invariance. Again, the
sharp range of estimates is obtained, but the results are limited to the case of 2-dimensional
cones.
Finally, Chapter 4 discusses some recent joint work of the author and Jim Wright considering
the restriction problem over rings of integers modulo a prime power. The sharp range of
estimates is obtained for Fourier restriction to the moment curve in finitely-generated free
modules over such rings. This is achieved by lifting the problem to the p-adics and applying
a classical argument of Drury in this setting. This work aims to demonstrate that rings of
integers offer a simplified model for the Euclidean restriction problem.