Abstract
Letf=g t+h t be the optimal decomposition for calculating the exact value of theK-functionalK(t, f;\(\bar X\)) of an elementf with respect to a couple\(\bar X\) =(X 0 ,X 1) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spacesX 0 andX 1 is eitherL ∞ orL 1. Many examples are given of couples of lattices\(\bar X\) for which |g t| increases monotonically a.e. with respect tot. It is shown that this property implies a sharpened estimate from above for the Brudnyi-KrugljakK-divisibility constant γ(\(\bar X\)) for the couple. But it is also shown that certain couples\(\bar X\) do not have this property. These also provide examples of couples of lattices for which γ(\(\bar X\)).
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Research supported by the Technion V. P. R. Fund.
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Cwikel, M., Keich, U. Optimal decompositions for theK-functional for a couple of Banach lattices. Ark. Mat. 39, 27–64 (2001). https://doi.org/10.1007/BF02388790
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DOI: https://doi.org/10.1007/BF02388790