Joint spectral multipliers for mixed systems of operators

TL;DR

This paper establishes a Marcinkiewicz-type multiplier theorem for mixed systems of commuting operators, including applications to Ornstein-Uhlenbeck and Gaussian-bound operators, with results on weak type bounds for Laplace transform multipliers.

Contribution

It introduces a general multiplier theorem for mixed operator systems with different functional calculus properties, extending previous results to new classes of operators.

Findings

Proved a Marcinkiewicz-type multiplier theorem for mixed systems of operators.

Established weak type (1,1) bounds for Laplace transform type multipliers.

Included results on Riesz transforms involving Ornstein-Uhlenbeck and Gaussian-bound operators.

Abstract

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{L}+A)^{-1}$ and $\mathcal{L}(\mathcal{L}+A)^{-1}.$