Boundary limits for bounded quasiregular mappings

Abstract

In this paper we establish results on the existence of nontangential limits for weighted $\Cal A$-harmonic functions in the weighted Sobolev space $W_w^{1,q}(\Bbb B^n)$, for some $q>1$ and $w$ in the Muckenhoupt $A_q$ class, where $\Bbb B^n$ is the unit ball in $\Bbb R^n$. These results generalize the ones in section \S3 of [KMV], where the weight was identically equal to one. Weighted $\Cal A$-harmonic functions are weak solutions of the partial differential equation $$\text{div}(\Cal A(x,\nabla u))=0,$$ where $\alpha w(x) |\xi|^{q} \le < \Cal A(x,\xi),\xi >\le \beta w(x) |\xi|^{q}$ for some fixed $q\in (1,\infty)$, where $0<\alpha\leq \beta<\infty$, and $w(x)$ is a $q$-admissible weight as in Chapter 1 in [HKM]. Later, we apply these results to improve on results of Koskela, Manfredi and Villamor [KMV] and Martio and Srebro [MS] on the existence of radial limits for bounded quasiregular mappings in the unit ball of $\Bbb R^n$ with some growth restriction on their multiplicity function.