A fractional order Hardy inequality
Let \(D\subset \mathbb R^d\), \(d\in \mathbb N\), be a domain such that \(D^c= \mathbb R^d\setminus D \not = \emptyset\); put \(\delta_D (x) = \text{dist}\,(x,D^c)\) for \(x\in D\). Assume that \(\alpha, p >0\). The main result of the paper reads as follows: The inequality \[ \int_D \frac{| u(x)| ^p}{\delta_D(x)^{\alpha}}\,dx \leq c \int_D\int_D \frac{| u(x) - u(y)| ^p}{| x-y| ^{d+\alpha}} \,dx\,dy \] holds for all \(u\in C_c(D)\) (with a finite constant \(c = c(D,\alpha,d,p)\)) in each of the following cases: (1) \(D\) is a bounded Lipschitz domain and \(\alpha >1\); (2) \(D\) is a complement of a bounded Lipschitz domain and \(\alpha\notin \{1,d\}\); (3) \(D= \mathbb R^d\setminus \{x_0\}\), where \(x_0 \in \mathbb R^d\), and \(\alpha \not= d\); (4) \(D\) is a domain above the graph of a Lipschitz function \(f: \mathbb R^{d-1} \rightarrow \mathbb R\) and \(\alpha \not = d\).
scientific article; zbMATH DE number 2138875