Lipšicov prostor i kvazikonformna preslikavanja

Abstract

This thesis has been written under the supervision of my mentor Prof. Miodrag Mateljevi c, and my co-mentor dr. Vladimir Bozin at the University of Belgrade in the academic year 2012-2013. The topic of this thesis is Complex analysis related with geometric function theory, more precisely the theory of quasiconformal mappings in the Euclidean n-dimensional space. For good survey of the eld, see F. W. Gehring [20] in the handbook of K uhnau [33] which also contains many other surveys on quasiconformal mappings and related topics. The main source in this dissertation is J. V ais al a [67]. The thesis is divided into three chapters. Chapter 1 is divided into 5 sections. In this chapter, we focus on quasiconformal mappings in Rn and discuss various equivalent denitions. We give The Modulus of family of curves in the rst section, geometric denition of quasiconformal space mappings in second section, analytic denition of quasiconformal space mappings in third section, equivalence of the denitions in fourth section, and the Beltrami equation in fth section. Chapter 2 is divided into 5 sections. We begin by generalizing the class of Lip( ), 0 < 1, and some properties of that class. Chapter 2 is devoted to understanding the properties by introducing the notion of Linearity, Dierentiability, and majorants. A majorant function is a certain generalization of the power functions t, this is done in the rst section. In the second section we introducing the notion of moduli of continuity with its Some Properties which gotten from I.M. Kolodiy, F. Hildebrand paper [39]. In third section we produced harmonic mapping as preliminary for the fourth section which including subharmonicity of jfjq of harmonic quasiregular mapping in space. In the last section we introducing estimation of the Poisson kernel which were extracted from Krantz paper [42]. Chapter 3 is divided into 3 sections. This chapter is include the main result in this dissertation. In this chapter we prove that !u() C!f (), where u : ! Rn is the harmonic extension of a continuous map f : @ ! Rn, if u is a K-quasiregular map and is bounded in Rn with C2 boundary. Here C is a constant depending only on n, !f and K and !h denotes the modulus of continuity of h. We also prove a version of this result for !-extension domains with c-uniformly perfect boundary and quasiconformal mappings, and we state some results regarding HQC self maps of the quadrant Q = fz : z = x + iy; x; y > 0g.