Weighted sobolev-poincaré type inequalities

This thesis consists of six chapters. In Chapter one, general information is provided about the topics covered in the thesis. Chapter two presents the fundamental concepts to be used throughout the thesis. In Chapter three, the materials and methods employed in the thesis are discussed. The first part of Chapter four introduces the definition of the Ap Muckenhoupt condition and some of its properties. Additionally, weighted inequalities for the maximal operator and the Riesz potential are examined. In the second part, the weighted Sobolev–Poincaré inequality is analyzed. Subsequently, the Hölder continuity of solutions to elliptic and degenerate elliptic equations is investigated. In the third part, the weighted Sobolev–Poincaré inequality is studied for solutions to parabolic equations. Chapter five is devoted to the conclusion. Finally, Chapter six includes the references. Keywords: Weighted inequalities, degenerate elliptic equations, weighted Sobolev-Poincaré type inequalities, parabolic weighted Sobolev-Poincaré type inequalities