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Strong \(A_ \infty\) weights, Sobolev inequalities and quasiconformal mappings. (English) Zbl 0752.46014
Analysis and partial differential equations, Coll. Pap. dedic. Mischa Cotlar, Lect. Notes Pure Appl. Math. 122, 101-111 (1990).

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A locally integrable function \(w\) on \(\mathbb{R}^ n\) is called an \(A_ \infty\) weight if for each \(\varepsilon>0\) there is a \(\delta>0\) such that if \(Q\subseteq\mathbb{R}^ n\) is any cube and \(E\subseteq Q\) satisfies \(| E|\leq\delta| Q|\) then \(w(E)\leq\varepsilon w(Q)\). Here \(| E|\) denotes the Lebesgue measure of \(E\) and \(w(E)=\int_ E w(x)dx\). An \(A_ \infty\) weight induces in a natural way a measure distance function \(\delta(x,y)\) and a geodesic distance function \(d(x,y)\). An \(A_ \infty\) weight is strongly \(A_ \infty\) when these two distances are equivalent. For example, the Jacobian \(w\) of a quasiconformal homeomorphism on \(\mathbb{R}^ n\) is always strongly \(A_ \infty\). For such a \(w\) one obtains the Sobolev inequality \[ (\int_{\mathbb{R}^ n}| g(x)|^ q w(x)dx)^{1/q}\leq C(\int_{\mathbb{R}^ n} [w(x)^{-1/n}|\nabla g(x)|]^ p w(x)dx)^{1/p}, \] where \(1\leq p<n\), \({1\over q}={1\over p}-{1\over n}\), and also the Poincaré inequality \[ w(B)^{-{n+1\over n}}\int_ B \int_ B | g(x)-g(y)| w(x)w(y)dx dy\leq c\int_{2B}(w(x)^{- 1/n}|\nabla g(x)|)w(x)dx \] for \(C^ 1\) functions \(g\) with compact support support. The main result is that the Sobolev and Poincaré inequalities hold for any strongly \(A_ \infty\) weight.
Reviewer: A.Pryde (Clayton)

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems