Weighted norm estimates and Lp-spectral independence of linear operators

We investigate the Lp-spectrum of linear operators defined consistently on Lp(Ω) for p0 ≤ p ≤ p1, where (Ω, μ) is an arbitrary σ-finite measure space and 1 ≤ p0 < p1 ≤ ∞. 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 (Ω, μ); the balls with respect to this semi-metric are required to satisfy a subexponential 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 elliptic operators in Euclidean space and generators of semigroups that satisfy (generalized) Gaussian bounds — to indicate improvements. AMS subject classification: 45P05, 35P05, 47A10, 47D03. keywords: Lp-spectrum, weighted norm estimates, integral operators, resolvent, elliptic operators, heat kernel estimates. ∗supported by Deutsche Forschungsgemeinschaft