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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).
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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems