Some Regularity Results for Certain Weakly Quasiregular Mappings on the Heisenberg Group and Elliptic Equations

Abstract

The thesis is organized as follows. In chapter 1, we set up a higher integrability result for the horizontal part of certain weakly quasiregular maps on the Heisenberg group. Unlike the Euclidean case, the exponential of the integrability is not near the homogeneous dimension Q that is not analogous to the Euclidean setting. Chapter 2 is devoted to the study of self-improving regularity for certain subelliptic equations. The difficulty of this problem in the Carnot group is that the Whitney extension theorem and the main result in the Carnot group can be obtained only for fourth-order homogeneous subelliptic systems from the arguments in (J. L. Lewis, On the very weak solutions of certain elliptic systems, Comm. Part. Diff. Equ., 18 (1993), 1515-1537.). Since the p-sub-Laplace equation is a very special case of the nonlinear subelliptic equations we can establish a better result in this case via the arguments from (R. R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.). Chapter 3 provides a discussion of selfimproving regularity for the degenerate elliptic equations in the Euclidean space. The main result of Chapter 3 extends a result of Lewis from (J. L. Lewis, On the very weak solutions of certain elliptic systems, Comm. Part. Diff. Equ., 18 (1993), 1515-1537.) to the degenerate elliptic systems. The proof relies on the weighted pointwise Sobolev inequality for higher order derivatives which is a useful tool in study of higher order degenerate elliptic systems.