Vector--valued functional calculus for a convolution algebra of distributions on cone

For the Fourier image $widehat{D}'_Gamma$ of the algebra $D'_Gamma$ of the distributions with supports on a cone $Gamma$ the functional calculus for generators of $n$-parametric $(C_o)$-semigroups of operators is determined. For this purpose, we consider construction of the dual pair $langle widehat{D}'_Gamma, widehat{D}_Gammarangle$, and provide some examples with respect to the formula of operator calculus.