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L\({}^ p\)-averaging domains and the Poincaré inequality. (English) Zbl 0706.26010

Let D be a proper subdomain of \({\mathbb{R}}^ n\) and \(u\in L^ 1_{loc}(D)\). If \(A\subset D\) with \(m(A)<\infty\) and \(p\geq 1\), then write \(\| u\|_{A,p}=((1/m(A))\int_{A}| u-u_ A|^ pdm)^{1/p};\) here \(u_ A\) is the average of u in A. Now u is said to be in BMO(D) if \(\| u\|_{*,D}=\sup_{B\subset D}\| u\|_{B,1}<\infty;\) here B is any ball. The main part of the paper is devoted to study \(L^ p\)-averaging domains: There are domains D with \(m(D)<\infty\) such that for some \(c<\infty\), \(\| u\|_{D,p}\leq c \sup \| u\|_{B,p}\) over all balls \(B\subset D\) and all \(u\in L^ 1_{loc}(D).\) The \(L^ p\)-averaging domains can be characterized in terms of the quasi-hyperbolic metric \(k_ D(x,y)\) of D; \(k_ D(x,y)=\inf_{\gamma}\int_{\gamma}d(z,\partial D)^{-1}ds,\) where \(\gamma\) is any rectifiable curve in D joining x to y. In fact, D is an \(L^ p\)-averaging domain iff \(k_ D(\cdot,x_ 0)\in L^ p(D)\) for some \(x_ 0\in D\). Now it is easy to show that \(L^ p\)-averaging domains include John domains. \(L^ p\)-averaging domains have applications to other classes of domains: for \(p\geq n\) it is shown that in an \(L^ p\)-averaging domain the Poincaré inequality holds for functions \(u\in W^{1,p}(D).\) The author also studies similar questions when \(\| u\|_{A,p}\) is replaced by \(osc_ A u=\sup_{A} u- \inf_{A}u;\) D is an oscillation domain if \(osc_ D u\leq c \sup osc_ B u\) for all balls \(B\subset D\) and all functions u. These domains can be characterized by a chaining property involving a finite number of balls. For example: If each point in D lies in a ball \(B\subset D\) of fixed radius \(\delta >0\), then D is an oscillation domain.
Reviewer: O.Martio

MSC:

26B35 Special properties of functions of several variables, Hölder conditions, etc.
26D10 Inequalities involving derivatives and differential and integral operators
42B05 Fourier series and coefficients in several variables
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