Generalized Hille–Phillips type functional calculus for multiparameter semigroups

For generators of $n$-parameter strongly continuous operator semigroups in a Banach space, we construct a Hille–Phillips type functional calculus, the symbol class of which consists of analytic functions from the image of the Laplace transform of the convolution algebra of temperate distributions supported by the positive cone $\mathbb R^n_+$. The image of such a calculus is described with the help of the commutant of the semigroup of shifts along the cone. The differential properties of the calculus and some examples are presented.