Optimal decompositions for the$K$-functional for a couple of Banach lattices

Michael Cwikel Department of Mathematics, Technion-Israel Institute of Technology Uri Keich Computer Science and Engineering Department, University of California, San Diego

TBD mathscidoc:1701.332951

Arkiv for Matematik, 39, (1), 27-64, 1999.9
Let$f=g$_{t}+$h$_{t}be the optimal decomposition for calculating the exact value of the$K$-functional$K(t, f$; $$\bar X$$ ) of an element$f$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 spaces$X$_{0}and$X$_{1}is either$L$^{∞}or$L$^{1}. Many examples are given of couples of lattices $$\bar X$$ for which |$g$_{t}| increases monotonically a.e. with respect to$t$. It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak$K$-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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  title={Optimal decompositions for the$K$-functional for a couple of Banach lattices},
  author={Michael Cwikel, and Uri Keich},
  booktitle={Arkiv for Matematik},
Michael Cwikel, and Uri Keich. Optimal decompositions for the$K$-functional for a couple of Banach lattices. 1999. Vol. 39. In Arkiv for Matematik. pp.27-64. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203600552824760.
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