Certain geometric maximal functions in harmonic analysis
dc.contributor.advisor
Hickman, Jonathan
dc.contributor.advisor
Wright, James
dc.contributor.author
Govindan Sheri, Aswin
dc.date.accessioned
2023-07-17T11:25:09Z
dc.date.available
2023-07-17T11:25:09Z
dc.date.issued
2023-07-17
dc.description.abstract
The broad theme of the thesis is of geometric maximal functions associated to
curved surfaces. We produce novel results about two maximal functions of different types, presented in two parts of the thesis.
In the first part (Chapter 2), we study the Lᵖ → Lᵖ boundedness of a lacunary
maximal function on a graded homogeneous group. The main theorem of this
part generalises the existing maximal results in specific homogeneous groups, such as the Euclidean space and the Heisenberg group. Using an iteration scheme, we estimate the maximal function, assuming that the measure associated to the maximal function satisfies a curvature condition.
This second part of this thesis (Chapters 3 and 4) deals with the problem of
Lᵖ → Lᵖ boundedness of a Nikodym maximal function in the Euclidean space.
The maximal function is defined using a one-parameter family of tubes in Rᵈ⁺¹, whose directions are determined by a non-degenerate curve in Rᵈ. These operators naturally arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for d = 2 and d = 3 to general dimensions.
en
dc.identifier.uri
https://hdl.handle.net/1842/40789
dc.identifier.uri
http://dx.doi.org/10.7488/era/3545
dc.language.iso
en
en
dc.publisher
The University of Edinburgh
en
dc.subject
harmonic analysis
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dc.subject
maximal functions
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dc.title
Certain geometric maximal functions in harmonic analysis
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dc.title.alternative
On certain geometric maximal functions in harmonic analysis
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dc.type
Thesis or Dissertation
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dc.type.qualificationlevel
Doctoral
en
dc.type.qualificationname
PhD Doctor of Philosophy
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