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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)

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