An estimate in the spirit of Poincaré's inequality
Let \(\Omega \subset \mathbb{R}^N\), \(N \geq 1\), be a bounded domain with Lipschitz boundary. If \(1 \leq p < \infty\), then it is known that there is a positive constant \(A = A (p, \Omega)\) such that the Poincaré inequality \[ \int_{\Omega} | f - f_{\Omega}| ^p \leq A \int_{\Omega} | Df| ^p \tag{1} \] holds for all \(f \in W^{1, p} (\Omega)\); here \(f_{\Omega} = | \Omega| ^{-1} \int_{\Omega} f\). Let \((\rho_n)_{n \in \mathbb{N}} \subset L^1 ( \mathbb{R}^n)\) be a sequence of radial functions satisfying \[ \begin{aligned} \rho_n \geq 0 \;\text{a.e.\;in }\mathbb{R}^n, \quad \int_{ \mathbb{R}^n} \rho_n = 1 &\quad \text{for all }n \in \mathbb{N},\\ \lim_{n \to \infty} \int_{| x| > \delta} \rho_n (x) \, dx = 0 &\quad \text{for all }\delta > 0. \end{aligned} \] The main result of the paper reads as follows. Let \(N \geq 2\). Given \(\delta > 0\), there is \(n_0 \geq 1\) sufficiently large such that, for all \(f \in L^p (\Omega)\) and \(n \geq n_0\), \[ \int_{\Omega} | f - f_{\Omega}| ^p \leq \biggl({A\over K_{p, N}} + \delta\biggr) \int_{\Omega} \int_{\Omega} {| f (x) - f (y)| ^p\over {| x - y| ^p}} \rho_n (| x - y| ) \, dx \, dy, \tag{2} \] where \(K_{p, N} = \oint_{S^{N - 1}} | e_1 \cdot \sigma| ^p \, {d{ H}} ^{N - 1}\). Note that (2) implies (1). To prove (2), the author makes use of the following compactness assertion: Let \(N \geq 2\). If \((f_n)_{n \in \mathbb{N}}\) is a bounded sequence satisfying \[ \int_{\Omega} \int_{\Omega} {| f_n (x) - f_n (y)| ^p\over | x - y| ^p} \rho_n (| x - y| ) \, dx \, dy \leq B \quad \text{for\;all} \;n \in \mathbb{N}, \tag{3} \] then \((f_n)\) is relatively compact in \(L^p (\Omega)\). Moreover, if \(f_{n_j} \to f\) in \(L^p (\Omega)\), then (i) \(f \in W^{1, p} (\Omega)\) when \(1 < p < \infty\); \noindent (ii) \(f \in B W (\Omega)\) when \(p = 1\). In both cases \(\int_{\Omega} | \nabla f| ^p \leq B/K_{p, N}\) with \(B\) from (3). If \(N = 1\), the above results hold under some additional assumption on the sequence~\((\rho_n)\).
scientific article; zbMATH DE number 2075132