The k_t--functional for the interpolation couple L^\infty(dμ;L^1(dν)), L^\infty(dν;L^1(dμ))

Abstract

Let $(M,\mu )$ and $(N,\nu )$ be measure spaces. In this paper, we study the $K_t$--\,functional for the couple $$A_0=L^{\infty }(d\mu ;L^1(d\nu )),A_1=L^{\infty }(d\nu ;L^1(d\mu )).$$ Here, and in what follows the vector valued $L^p$--\,spaces $L^p(d\mu ;L^q(d\nu ))$ are meant in Bochner's sense. One of our main results is the following, which can be viewed as a refinement of a lemma due to Varopoulos [V]. \proclaim Theorem 0.1. Let $(A_0,A_1)$ be as above. Then for all $f$ in $A_0+A_1$ we have $$\frac{1}{2}{K}_t(f;A_0,A_1)\le \sup \left\{\right(\mu (E)\vee t^{-1}\nu (F){)}^{-1}{\int }_{E\times F}|f|d\mu d\nu \}\le K_t(f;A_0,A_1),$$ where the supremum runs over all measurable subsets $E\subset M,F\subset N$ with positive and finite measure and $u\vee v$ denotes the maximum of the reals $u$ and $v$.