Multilinear interpolation theorems (Q2732518)

Let \((\Omega,\mu)\) be a measure space with \(\mu\) complete \(\sigma\)-finite, and let \(L^0(\mu)\) be the space of all equivalence classes of measurable functions on \(\Omega\) with the topology of convergence in measure on \(\mu\)-finite sets. A Banach lattice on \((\Omega,\mu)\) is a Banach lattice in \(L^0(\mu)\). If \(\overline{X}=(X_0,X_1)\) is a couple of Banach lattices on \((\Omega,\mu)\) and \(\phi\colon R_{+}\times R_{+}\to R_{+}\) is nondecreasing in each variable, homogeneous of degree one and concave, then the Calderón-Lozanovskii space \(\phi (\overline{X})\) consists of all \(x\in L^0(\mu)\) such that for some \(\lambda > 0\), \(|x|\leq\lambda\phi (|x_0|,|x_1|)\) \(\mu\)-a.e. for some \(x_j\in X_j\) with \(\|x_j\|_{X_j}\leq 1\), \(j=1,2\). The space \(\phi (\overline{X})\) is a Banach lattice on \((\Omega,\mu)\) equipped with the norm \(\|x\|\) equal to the infimum of all \(\lambda >0\) as above. NEWLINENEWLINENEWLINEIn the paper under review a necessary and sufficient condition is given for the interpolation theorem to hold for positive multilinear operators from the product of Caldéron-Lozanovskii spaces. In a particular case, if the parameters generating those spaces are the same concave functions, the condition is equivalent to supermultiplicativity of the generating function. This result is used to obtain interpolation theorems for the multilinear operators from the product of certain interpolation orbit spaces, in particular Ovchinnikov's lower method spaces, into generalized Marcinkiewicz spaces.