an:00902703
Zbl 0860.30018
Heinonen, Juha; Koskela, Pekka
Weighted Sobolev and Poincar?? inequalities and quasiregular mappings of polynomial type
EN
Math. Scand. 77, No. 2, 251-271 (1995).
00034259
1995
j
30C65 35J70 46E35
quasiregular mapping; weighted Sobolev inequality; weighted Poincar?? inequality; degenerate elliptic equations; strong \(A_ \infty\)-weight
Let \(f:\mathbb{R}^n\to\mathbb{R}^n\) be a quasiregular mapping. The authors prove that the weight \(w(x)=J(x,f)^{1-p/n}\) is \(p\)-admissible if and only if \(f\) is of polynomial type. The map \(f\) is said to be of polynomial type if \(|f(x)|\to\infty\). Recall also that a weight \(w(x)\) is \(p\)-admissible if it satisfies a weighted Sobolev inequality, a weighted Poincar?? inequality and a doubling condition and if the gradient is unique in the relevant weighted Sobolev space. (See the first author, \textit{T. Kilpelainen} and \textit{O. Martio}, Nonlinear potential theory of degenerate elliptic equations (1993; Zbl 0780.31001).) These four conditions are needed in order to use the Moser iteration method for degenerate elliptic equations. The admissibility of this particular class of weights settles a question of \textit{B. ??ksendal} [Comm. partial differential equations 15, 1447-1459 (1990; Zbl 0719.31002)].
The authors set forth by first characterizing quasiregular maps of polynomial type in terms of six equivalent conditions. We mention only three: (i) \(f\) is of polynomial type, (ii) \(J(x,f)\) is a doubling weight and (iii) \(J(x,f)\) is a strong \(A_\infty\)-weight. (See \textit{G. David} and \textit{S. Semmes}, Analysis and partial differential equations, Lect. Notes Pure Appl. Math. 122, 101-111 (1990; Zbl 0752.46014).) As an application of this result, they show that the image of a ball under a quasiregular map \(f:\mathbb{R}^n\to\mathbb{R}^n\) of polynomial type is a John domain. In their proofs of the Sobolev and Poincar?? inequality for the weight \(w(x)= J(x,f)^{1-p/n}\), the authors present a different more elementary approach. This proof centers on the strong \(A_\infty\)-weight characterization of the Jacobian of the quasiregular map.
S.Staples (Fort Worth)
Zbl 0780.31001; Zbl 0719.31002; Zbl 0752.46014