Chua, Seng-Kee Weighted Sobolev inequalities of mixed norm. (English) Zbl 0879.46009 Real Anal. Exch. 21(1995-96), No. 2, 555-571 (1996). Let \(1<p\leq q<\infty \), let \(R=I\times J\) be a parallelepiped, and let \(v,w\) be weights. X. Shi and 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). Reviewer: Luboš Pick (Praha) Cited in 1 ReviewCited in 2 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators Keywords:weighted Poincaré inequality; mixed norm; \(A_ p\)-weights; Boman chain condition; Lipschitz domain Citations:Zbl 0779.42011 × Cite Format Result Cite Review PDF