Boundary values of analytic functions on the disc

Abstract

In this bachelor's thesis we will solve the Dirichlet problem with an L^p(T) boundary function. First, we will focus on the holomorphic version of the Dirichlet problem and introduce Hardy space theory, from which will follow a sufficient condition on the Fourier coefficients of the boundary function. Then we will prove the Marcinkiewicz interpolation theorem. After that we introduce the conjugate function "tilde f", which equals the Hilbert transform of f, and use functional analysis to prove an important duality argument of the Hilbert transform. Finally, we will give several different proofs for the boundedness of the map f ↦ tilde f using the Marcinkiewicz interpolation theorem and the duality argument: the last proof will be done rigorously from scratch, i.e. without relying on (unproved) arguments from other literature.