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A Carlson type inequality with blocks and interpolation. (English) Zbl 0824.46088
Summary: An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function [see Yu. A. Brudnyj and the first author, ‘Interpolation functors and interpolation spaces. Vol. 1’, Amsterdam etc.: North-Holland Math. Libr. 47, XV, 718 p. (1991; Zbl 0743.46082), pp. 320-325]. Our Carlson type inequality is used to characterize Peetre’s interpolation functor \(\langle \rangle_ \varphi\) [see J. Peetre, Rend. Sem. Mat. Univ. Padova 46(1971), 173-190 (1972; Zbl 0233.46047)] and its Gagliardo closure on couples of functional Banach lattices in terms of the Calderón-Lozanovskij construction.
Our interest in this functor is inspired by the fact that if \(\varphi= t^ \theta\) \((0< \theta< 1)\), then, on couples of Banach lattices and their retracts, it coincides with the complex method [G. Ya. Lozanovskij, Probl. Mat. Anal. 7, 83-99 (1979; Zbl 0416.46021) and V. A. Sestakov, Vestnik Leningrad. Univ. 1974, No. 19 (Mat. Meh. Astron. No. 4), 64-68 (1974; Zbl 0297.46028)] and, thus, it may be regarded as a “real version” of the complex method.

MSC:
46M35 Abstract interpolation of topological vector spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D15 Inequalities for sums, series and integrals
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