## 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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