Joint Spectral Multipliers for Mixed Systems of Operators

Abstract

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators L=(L_1,ldots ,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},mathcal {L}(mathcal {L}+A)^{-1}.