Interpolation for analytic families of multilinear operators on metric measure spaces
Volume 267 / 2022
Abstract
Let $(X_j, d_j, \mu _j)$, $j=0,1,\ldots , m$, be metric measure spaces. Given $0 \lt p^\kappa \le \infty $ for $\kappa = 1, \ldots , m$, and an analytic family of multilinear operators $$ T_z: L^{p^1}(X_1)\times \cdots \times L^{p^m}(X_m) \to L^1_{\rm loc}(X_0) $$ 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 by Grafakos and Mastyło (2014). 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 on an auxiliary parameter $z$. An application of the main theorem concerning bilinear estimates for Schrödinger operators on $L^p$ is included.
