Optimal decompositions for the \(K\)-functional for a couple of Banach lattices
Let \(X_0\) and \(X_1\) be Banach lattices of (equivalence classes of) real valued measurable functions on the same measure space \((\Omega,\Sigma,\mu)\). It is known that \(X_0\) and \(X_1\) form a Banach couple \(\overline{X}=(X_0,X_1)\) in the sense of interpolation theory. Let \(\{f=g_t+h_t\}_{t>0}\) be the optimal decomposition for calculating the exact value of the \(K-\)functional \(K(t,f,\overline{X})\) of an element \(f\) with respect to a couple \(\overline{X}= (X_0,X_1)\) of Banach lattices of measurable functions. In this paper the authors show that this decomposition has a rather simple form in many cases where one of the spaces \(X_0\) and \(X_1\) is either \(L^{\infty}\) or \(L^1\). Many examples are given of couples of lattices \(\overline{X}\) for which \(|g_t|\) increases monotonically a.e. wich respect to \(t\). It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak \(K-\)divisibility constant \(\gamma(\overline{X})\) for the couple. It is also shown that certain couples \(\overline{X}\) do not have this property. These also provide examples of couples of lattices for which \(\gamma(\overline{X})>1\).
scientific article; zbMATH DE number 1987643