Weighted norm estimates and Lp-spectral independence of linear operators

Abstract

We investigate the Lp-spectrum of linear operators defined consistently on Lp() for p0 � pp1, where (,µ) is an arbitrary �-finite measure space and 1 � p0 < p1 � 1. We prove p-independence of the Lp-spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (,µ); the balls with respect to this semi-metric are required to satisfy a subexponen- tial volume growth condition. We show how previous results on Lp-spectral independence can be treated as special cases of our results and give examples—including strictly ellip- tic operators in Euclidean space and generators of semigroups that satisfy (generalized) Gaussian bounds—to indicate improvements.