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Imbedding theorems for spaces of UMD-valued functions. (English. Russian original) Zbl 0836.46022
Russ. Acad. Sci., Dokl., Math. 47, No. 2, 274-277 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 329, No. 4, 408-410 (1993).
Imbedding theorems are presented for anisotropic Sobolev spaces \(W_p^\ell (G, X)\), where \(\ell=(\ell_1, \dots\) \(\dots,\ell_n)\) is an \(n\)-tuple of non-negative integers, and for anisotropic Liouville (potential) spaces \(L^r_p (\mathbb{R}^n, X)\) with \(r= (r_1, \dots, r_n)>0\). \(G\) is assumed to satisfy the \(\ell\)-horn condition and \(X\) is an UMD space, that is, there is a function \(\xi\) on \(X\times X\), symmetric and convex in each variable such that \(|a+b |_X \geq \xi( a,b)\) \(\forall |a|_X= |b|_X =1\). An extension theorem is stated for \(W_p^\ell (G, X)\) provided \(X\) has an unconditional basis. Imbedding theorems for \(L^r_p (\mathbb{R}^n, X)\) as well as a Mikhlin type multiplier theorem are presented for \(X\) being a Banach lattice.
Reviewer: M.Krbec (Praha)

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B42 Banach lattices