The \(L_{\nu }^{(\rho)}\)- transformation on McBride's spaces of generalized functions (Q2713582)

The authors consider a generalized Laplace transformation \(L^{(\rho)}_{\nu } (\nu \in \mathbb C, \rho >0)\), introduced by \textit{E. Krätzel} [Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-Naturw. Reihe 14, 369-381 (1965; Zbl 0168.10204)]. They show that it holds NEWLINE\[NEWLINE \int ^{\infty }_0 (L^{(\rho)}_{\nu } \varphi) (x) \psi (x) dx = \int ^{\infty }_0 \varphi (x) (L^{(\rho)}_{\nu } \psi)(x) dx NEWLINE\]NEWLINE for \(\varphi \in F_{p,\mu }\), \(\psi \in F_{p',\mu '}\), if some conditions on \(\nu ,\rho ,p,\mu \) are fulfilled; here \(p',\mu '\) are defined by relations \(1/p + 1/p' = 1\), \(\mu - \mu ' = 1/p-1/p'\), and \(F_{p,\mu }\) \((p\geq 1, \mu \in \mathbb C)\) denotes a test function space, introduced by \textit{A. C.~McBride} [Fractional calculus and integral transforms of generalized functions (1979; Zbl 0423.46029)] (which is a subset of \(C^{\infty }(\mathbb R^+)\)). This formula allows to extend the operator \(L^{(\rho)}_{\nu }\) onto the corresponding distribution spaces. NEWLINENEWLINENEWLINEThe authors prove also a result on a composition of this extended operator \(L^{(\rho)}_{\nu }\) with a differential operator.