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Weighted Sobolev inequalities of mixed norm. (English) Zbl 0879.46009

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).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators

Citations:

Zbl 0779.42011