Sharp inequalities via truncation
In what follows \(X\) will be a metric space (with metric \(d)\) and \(\mu\) will be a Borel measure on \(X.\) Let \(\Omega\subset X\) be a domain with \(\mu (\Omega)<\infty.\) Fix \(p,q\in(0,\infty)\) and assume that the Sobolev-Poincaré inequality \[ \inf_{c\in{\mathbb R}} \left( \int_{\Omega}\left|u(y)-c\right|^{q}d\mu(y)\right) ^{1/q}\leq C_{0}\left( \int_{\Omega}g^{p}(y) d\mu (y)\right) ^{1/p} \] is stable under truncations, i.e., for every \(b\in{\mathbb R},\) \(0<t_{1} <t_{2}<\infty\) and \(\alpha\in\{-1,1\}\) the pair \(u_{t_{1},t_{2}},\) \(g_{t_{1},t_{2}}\) also satisfies the inequality above. Here \(u_{t_{1},t_{2}} =\min\{\max\{0,\alpha(u-b)-t_{1}\},t_{2}-t_{1}\}\) and \(g_{t_{1},t_{2}} =g\chi_{\{t_{1}<\alpha(u-b)\leq t_{2}\}}.\) Theorem 1 (i) asserts the existence of a constant \(C=C(p,q)\) such that \[ \inf_{c\in{\mathbb R}} \int_{0}^{\infty}t^{p-1}[\mu\left( \{x\in \Omega;\;\left|u(x)-c\right|>t\}\right) ]^{p/q} dt\leq CC_{0} ^{p}\int_{\Omega}g^{p}(y) d\mu(y). \] Fix \(s\in(1,\infty)\) and assume that the inequality \[ \inf_{c\in{\mathbb R}} \int_{\Omega}\exp\left( \frac{\left|u(y)-c\right|}{C_{1}\left\|g\right\|_{L^{s}(\Omega)}}\right) ^{s/(s-1)}d\mu(y)\leq C_{2}\mu(\Omega) \] is stable under truncations. Theorem 1 (ii) asserts the existence of a constant \(C=(s,\) \(C_{2})\) such that \[ \inf_{c\in{\mathbb R}} \int_{0}^{\infty}\frac{t^{s-1}}{\log^{s-1}\left( \frac{e\mu(\Omega)}{\mu\left( \{x\in\Omega;\;\left|u(x)-c\right|>t\}\right) }\right) } dt\leq CC_{1}^{s}\int_{\Omega}g^{s}(y) d\mu(y). \] The idea of using truncation in connection with Sobolev type inequalities can be traced to \textit{V. G. Maz'ya} [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 1145-1172 (1964; Zbl 0148.35602)]. Theorem 1 (ii) was inspired by previous work of \textit{N. S. Trudinger} [J. Math. Mech. 17, 473-483 (1967; Zbl 0163.36402)]. The conclusions (i) \& (ii) of Theorem 1 were known in the Euclidean setting when the boundary of \(\Omega\subset {\mathbb R}^{n}\) is sufficiently smooth and \(1\leq p<n\) and \(q=pn/(n-p)\) or \(s=n\). What makes this theorem interesting (even in this particular case) is that no assumption on the geometry of \(\Omega\) is made, except for the above first two inequalities.
scientific article; zbMATH DE number 1915712
scientific article