Prostori harmonijskih funkcija i harmonijska kvazikonformna preslikavanja

Abstract

This thesis has been written under the supervision of my mentor, Prof. dr. Miloš Arsenović at the University of Belgrade academic, and my co-mentor dr. Vladimir Božin in year 2013. The thesis consists of three chapters. In the first chapter we start from defnition of harmonic functions (by mean value property) and give some of their properties. This leads to a brief discussion of homogeneous harmonic polynomials, and we also introduce subharmonic functions and subharmonic behaviour, which we need later. In the second chapter we present a simple derivation of the explicit formula for the harmonic Bergman reproducing kernel on the ball in euclidean space and give a proof that the harmonic Bergman projection is Lp bounded, for 1 < p < 1, we furthermore discuss duality results. We then extend some of our previous discussion to the weighted Bergman spaces. In the last chapter, we investigate the Bergman space for harmonic functions bp, 0 < p < 1 on RnnZn. In the planar case we prove that bp 6= f0g for all 0 < p < 1. Finally we prove the main result of this thesis bq c bp for n=(k + 1) < q < p < n=k, (k = 1; 2; :::). This chapter is based mainly on the published paper [44]. M. Arsenović, D. Kečkić,[5] gave analogous results for analytic functions in the planar case. In the plane the logarithmic function log jxj, plays a central role because it makes a diference between analytic and harmonic case, but in the space the function /x/2-n; n > 2 hints at the contrast between harmonic function in the plane and in higher dimensions.