Date of Graduation

5-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Raich, Andrew

Committee Member

Harrington, Phillip S.

Second Committee Member

Bradshaw, Zachary

Keywords

Atomic Decomposition; Boundary Values in the Sense of Distributions; Hardy Spaces; Meta-Analytic Functions; Nonhomogeneous Cauchy-Riemann Equations; Schwarz Boundary Value Problem

Abstract

We prove that many of the boundary properties associated with functions in the classic holomorphic Hardy spaces on the complex unit disk are present in Hardy classes of solutions to certain nonhomogeneous Cauchy-Riemann equations and higher-order generalizations of these equations. Also, we explicitly solve generalizations of the Schwarz boundary value problem on the complex unit disk and the upper-half plane when the boundary condition is in terms of boundary values in the sense of distributions.

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