Joint spectral multipliers for mixed systems of operators

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