Weighted Hardy inequalities and the size of the boundary (Q958183)

This paper provides necessary and sufficient conditions for a domain \(\Omega\subset\mathbb{R}^{n}\) to admit the \((p,\beta)\)-Hardy inequality \[ \int_{\Omega}|u|^{p}d(x,\partial\Omega)^{\beta-p}\,dx\leq C\int_{\Omega}|\nabla u|^{p}d(x,\partial\Omega)^{\beta }\,dx \] for all \(u\in C_{0}^{\infty}(\Omega)\). The inequalities of this kind originate from the one-dimensional considerations of \textit{G.\,H.\thinspace Hardy, J.\,E.\thinspace Littlewood} and \textit{G.\,Pólya} in their book [``Inequalities'' (Cambridge Univ.\ Press) (\({}^1\)1934; JFM 60.0169.01, Zbl 0010.10703) (\({}^2\)1952; Zbl 0047.05302) (1988; Zbl 0634.26008)]. In the paper under review, it is proved that if a domain \(\Omega\) admits the \((p,\beta)\)-Hardy inequality (with \(1<p<\infty,\) \(\beta\neq p),\) then there exists \(\varepsilon=\varepsilon (C,p,\beta,n)>0\) such that, for each ball \(B\subset\) \(\mathbb{R}^{n}\), either \(\dim_{\mathcal{H}}(4B\cap(\mathbb{R}^{n}\backslash\Omega))>n-p+\beta +\varepsilon\) or \(\dim_{\mathcal{A}}(4B\cap(\mathbb{R}^{n}\backslash \Omega))>n-p+\beta-\varepsilon.\) This represents the weighted extension of a result first noticed by \textit{P.\,Koskela} and \textit{X.\,Zhong} [``Hardy's inequality and the boundary size'', Proc.\ Am.\ Math.\ Soc.\ 131, No.\,4, 1151--1158 (2003; Zbl 1018.26008)]. The sufficient conditions for the weighted Hardy inequalities exploit the size and geometry of the boundary. For example, it is shown that every unbounded John domain admits the \((p,\beta)\)-Hardy inequality for \(1<p<\infty\) and \(\beta\in\mathbb{R}\) provided that \(\dim_{\mathcal{A} }(\partial\Omega)<n-p+\beta\). Much more general sufficient conditions are indicated in Theorem 3.1 and Theorem 4.3 (respectively, for bounded and unbounded domains).