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

Abstract

Letf=g_t+h_t be the optimal decomposition for calculating the exact value of the K-functionalK(t, f; $\bar X$ ) of an element f with respect to a couple $\bar X$ =(X₀ , X₁) 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 X₀ and X₁ is either L^∞ or L¹. 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-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( $\bar X$ ).