Some Aspects of Weighted Kernel Functions on Planar Domains

Includes bibliographical refrences

PhD;2023;Mathematics

The broad aim of this thesis is to study some aspects of weighted kernel functions. In particular, this thesis has been driven by three principal goals: First, to study the boundary behaviour of weighted Bergman kernels. Second, to study holomorphic isometries between domains equipped with weighted Bergman metrics. Third, to study and investigate the interplay between the weighted classical objects of potential theory and conformal mapping. For a planar domain D and an admissible weight function on it, some aspects of the boundary behaviour of the corresponding weighted Bergman kernel are studied. First, under the assumption that the weight extends continuously to a smooth boundary point p of D and is non-vanishing there, we obtain a precise relation between the weighted Bergman kernel and the classical Bergman kernel near p. Second, when viewed as functions of such weights, the weighted Bergman kernel is shown to have a suitable additive and multiplicative property near such boundary points. We then study isometries with respect to the weighted Bergman metric. In particular, motivated by work of Mok, Ng, Chan--Yuan and Chan--Xiao--Yuan among others, we study the nature of holomorphic isometries from the unit disc with respect to the weighted Bergman metrics arising from weights of the form K^{-d}, where K is the classical Bergman kernel and d is a non-negative integer. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied in detail include those in which the isometry takes values in polydiscs, and the cartesian product of a unit disc and a ball, where each factor admits a weighted Bergman metric as above for possibly different non-negative integers d. Finally, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented. Finally, we study properties of weighted Szeg\H{o} and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell's work, the starting point is a weighted Kerzman--Stein formula that yields boundary smoothness of the weighted Szeg\H{o} kernel. This provides information on the dependence of the weighted Szeg\H{o} kernel as a function of the weight. When the weights are close to the constant function 1 (which corresponds to the unweighted case), it is shown that some properties of the unweighted Szeg\H{o} kernel propagate to the weighted Szeg\H{o} kernel as well. Finally, it is shown that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szeg\H{o} kernels and their conjugates, thereby extending Bell's list of kernel functions that are made up of simpler building blocks that involve the Szeg\H{o} kernel.