Some Resolvent Estimates in Harmonic Analysis

Abstract: This thesis contains three papers about three different estimates of resolvents in harmonic analysis. These papers are: Paper 1. ``A Wiener tauberian theorem for weighted convolution algebras of zonal functions on the automorphism group of the unit disc'' Paper 2. ``Uniform spectral radius and compact Gelfand transform'' Paper 3. ``Decomposable extension of the Cesáro operator on the weighted Bergman space and Bishop's property (b )'' The first paper concerns the classical resolvent transform for a commutative convolution algebra and its applications to tauberian theorems. The second paper concerns uniform estimates of resolvents and of inverses in commutative Banach (and quasi-Banach) algebras, in particular when the Gelfand transform is compact. In the last paper we consider the Cesáro operator and its action on weighted Bergman spaces. Using classical analysis we calculate the spectrum, produce estimates the resolvent and of its left inverse. The results are then used to retrieve operator theoretic information of the Cesáro operator on the weighted Bergman space.