Interpolation for analytic families of multilinear operators on metric measure spaces
Language
en
Document de travail - Pré-publication
English Abstract
Let (X j , d j , µ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p κ ≤ ∞ for κ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) × • • • L p m (X m) → L 1 loc (X 0), for z in the ...Read more >
Let (X j , d j , µ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p κ ≤ ∞ for κ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) × • • • L p m (X m) → 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 Schrödinger operators on L p is included.Read less <
English Keywords
multilinear operators
analytic families of operators
interpolation
bilinear estimates for Schrödinger operators
ANR Project
Analyse Réelle et Géométrie - ANR-18-CE40-0012
Origin
Hal imported