Spectral multipliers via resolvent type estimates on non-homogeneous metric measure spaces

TL;DR

This paper introduces a flexible technique for deriving spectral multiplier results for operators with finite propagation speed, applicable even in non-homogeneous spaces lacking doubling conditions, with implications for various differential operators.

Contribution

The paper presents a novel approach linking resolvent estimates to spectral multipliers, extending applicability to non-doubling spaces and operators with unbounded semigroups.

Findings

Spectral multipliers derived from resolvent estimates for operators with finite propagation speed.

Applicable to non-homogeneous metric measure spaces without doubling condition.

Established $L^p$ spectrum independence for certain differential operators.

Abstract

We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator. We show that, in this setting, spectral multipliers follow from resolvent type estimates. The most notable point of the paper is that our approach is very flexible and can be applied even if the corresponding ambient space does not satisfy the doubling condition or if the semigroup generated by an operator is not uniformly bounded. As a corollary we obtain the $L^p$ spectrum independence for several second order differential operators and recover some known results. Our examples include the Laplace-Belltrami operator on manifolds with ends and Schr\"odinger operators with strongly subcritical potentials.