Interpolation for analytic families of multilinear operators on metric measure spaces

TL;DR

This paper develops an interpolation theorem for analytic families of multilinear operators on metric measure spaces, extending classical results to more general settings with applications to Schrödinger operators.

Contribution

It introduces a new interpolation theorem for multilinear operators on metric measure spaces, relaxing pointwise conditions and including applications to bilinear estimates.

Findings

Established an interpolation theorem for analytic families of multilinear operators.

Proved the theorem applies to Schrödinger operators on L p spaces.

Extended classical complex interpolation results to metric measure space settings.

Abstract

Let (X j , d j , $\mu$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $\kappa$ $\le$ $\infty$ for $\kappa$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $\bullet$ $\bullet$ $\bullet$ L p m (X m) $\rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Stein's complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators T z are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schr{\"o}dinger operators on L p is included.