Weighted Sobolev inequalities of mixed norm
Let \(1<p\leq q<\infty \), let \(R=I\times J\) be a parallelepiped, and let \(v,w\) be weights. \textit{X. Shi} and \textit{A. Torchinsky} [Proc. Am. Math. Soc. 118, No. 4, 1117-1124 (1993; Zbl 0779.42011)] gave a sufficient condition on \(v, w\) in order that the Poincaré inequality \[ w(R)^{-1/q}|f-f_R|_{q(w)}\leq C|I|^{1/n}v(R)^{-1/p}|\nabla_1f|_{p(v)}+ C|J|^{1/m}v(R)^{-1/p}|\nabla_2f|_{p(v)} \] holds. The author generalizes this result to mixed norm inequalities on parallelepipeds and also to inequalities on products of domains satisfying the Boman chain condition (recall that this class contains for example Lipschitz domains).
scientific article; zbMATH DE number 1095847
scientific article