## The Hardy constant.(English)Zbl 0857.26005

Let $$\Omega$$ be either a bounded region in $$\mathbb{R}^N$$ or an incomplete $$N$$-dimensional Riemannian manifold of finite diameter. Suppose that the completion $$\overline\Omega$$ of $$\Omega$$ is compact and let $$\partial\Omega:=\overline\Omega\backslash\Omega$$. Let $$\rho(x,y)$$ denote the extension of the Riemannian distance function to $$\overline\Omega\times\overline\Omega$$ and $$d(x):=\min\{\rho(x,y);y\in \partial\Omega\}$$, $$x\in\Omega$$.
The author assumes that the inequality $\int_\Omega{|f(x)|^2\over d(x)^2} d\text{ vol}\leq c\int_\Omega |\nabla f(x)|^2 d\text{ vol}+ a\int_\Omega|f(x)|^2 d\text{ vol}\tag{$$*$$}$ holds for all $$f\in W^{1,2}_0(\Omega)$$ with some $$a,c\in[0,\infty)$$ and defines the weak Hardy constant of $$\Omega$$ to be the infimum of all $$c>0$$ such that there exists $$a<\infty$$ for which $$(*)$$ holds. Then he shows that every point of $$\Omega$$ has a local weak Hardy constant associated to it and that the weak Hardy constant is given by $$c=\max\{h(x);x\in \partial\Omega\}$$.
Moreover, the paper contains a number of methods of computing and estimating $$h(x)$$ for various types of boundary points.
Reviewer: B.Opic (Praha)

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators

### Keywords:

Hardy’s inequality; weak Hardy constant
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