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Weighted Sobolev and Poincaré inequalities and quasiregular mappings of polynomial type. (English) Zbl 0860.30018
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 Poincaré inequality and a doubling condition and if the gradient is unique in the relevant weighted Sobolev space. (See the first author, T. Kilpelainen and 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 B. Øksendal [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 G. David and 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 Poincaré 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.

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
35J70 Degenerate elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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