Generalized Hille-Phillips type functional calculus for multiparameter semigroups (Q467666)

Let \(A\) be the generator of an \(n\)-parameter strongly continuous semigroup on a Banach space \(E\). The authors construct a functional calculus for \(A\) of Hille-Phillips type by taking the Laplace transform of temperate Schwartz distributions supported by the positive cone \(\mathbb{R}_{+}^n\). The obtained operator \(\hat{f}(A)\) is unbounded on \(E\) and has dense domain. They also prove the differential properties of the calculus and present two examples: the \(n\)-parameter shift semigroup and the \(n\)-parameter Gaussian semigroup.