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L${}\sp p$-averaging domains and the Poincaré inequality. (English) Zbl 0706.26010
Let D be a proper subdomain of ${\bbfR}\sp n$ and $u\in L\sp 1\sb{loc}(D)$. If $A\subset D$ with $m(A)<\infty$ and $p\ge 1$, then write $\Vert u\Vert\sb{A,p}=((1/m(A))\int\sb{A}\vert u-u\sb A\vert\sp pdm)\sp{1/p};$ here $u\sb A$ is the average of u in A. Now u is said to be in BMO(D) if $\Vert u\Vert\sb{*,D}=\sup\sb{B\subset D}\Vert u\Vert\sb{B,1}<\infty;$ here B is any ball. The main part of the paper is devoted to study $L\sp p$-averaging domains: There are domains D with $m(D)<\infty$ such that for some $c<\infty$, $\Vert u\Vert\sb{D,p}\le c \sup \Vert u\Vert\sb{B,p}$ over all balls $B\subset D$ and all $u\in L\sp 1\sb{loc}(D).$ The $L\sp p$-averaging domains can be characterized in terms of the quasi-hyperbolic metric $k\sb D(x,y)$ of D; $k\sb D(x,y)=\inf\sb{\gamma}\int\sb{\gamma}d(z,\partial D)\sp{-1}ds,$ where $\gamma$ is any rectifiable curve in D joining x to y. In fact, D is an $L\sp p$-averaging domain iff $k\sb D(\cdot,x\sb 0)\in L\sp p(D)$ for some $x\sb 0\in D$. Now it is easy to show that $L\sp p$-averaging domains include John domains. $L\sp p$-averaging domains have applications to other classes of domains: for $p\ge n$ it is shown that in an $L\sp p$-averaging domain the Poincaré inequality holds for functions $u\in W\sp{1,p}(D).$ The author also studies similar questions when $\Vert u\Vert\sb{A,p}$ is replaced by $osc\sb A u=\sup\sb{A} u- \inf\sb{A}u;$ D is an oscillation domain if $osc\sb D u\le c \sup osc\sb B u$ 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

26B35Special properties of functions of several real variables, Hölder conditions, etc.
26D10Inequalities involving derivatives, differential and integral operators
42B05Fourier series and coefficients, several variables