Boundary Limits for Bounded Quasiregular Mappings

ISSN： 1050-6926

Source： Journal of Geometric Analysis, Vol.19, Iss.3, 2009-07, pp. : 708-718

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Abstract

In this paper we establish results on the existence of nontangential limits for weighted $mathcal{A}$ -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 Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755-766, 1996, where the weight was identically equal to one. Weighted $mathcal{A}$ -harmonic functions are weak solutions of the partial differential equation where $alpha w(x)|xi|^{q}le langle mathcal{A}(x,xi),xi ranglele beta w(x)|xi|^{q}$ for some fixed q∈(1,∞), where 0<α≤β<∞, and w(x) is a q-admissible weight as in Chap. 1 of Heinonen et al., Nonlinear Potential Theory, 2006.Later, we apply these results to improve on results of Koskela et al., Trans. Am. Math. Soc. 348(2), 755-766, 1996 and Martio and Srebro, Math. Scand. 85, 49-70, 1999 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.