Multidimensional Widder--Arendt theorem in locally convex spaces
Marko Kostic
TL;DR
This work extends the Widder-Arendt theorem to multidimensional vector-valued Laplace transforms in sequentially complete locally convex spaces. It proves that, under the Lipschitz-Radon-Nikodym property (and also in the integrated form for arbitrary locally convex spaces), differentiability and quantitative growth bounds of all mixed derivatives of a mv-valued function $F$ on $(\omega_1,\infty)\times\cdots\times(\omega_n,\infty)$ are equivalent to the existence of a locally integrable density $f$ representing $F$ as a multidimensional Laplace transform, with explicit exponential growth controls. The results yield analytic extension to the corresponding right half-planes and provide fractional-power representations and dual formulations via $x^*$-duality. The integrated form further relaxes assumptions and leverages convolution/Laplace-transform techniques to connect $F$ with $f$ in the dual pairing, highlighting the theory's breadth and potential impact on abstract ill-posed problems in locally convex settings.
Abstract
In this research article, we formulate and prove multidimensional Widder--Arendt theorem and integrated form of multidimensional Widder--Arendt theorem for functions with values in sequentially complete locally convex spaces. Established results seem to be new even for scalar-valued functions.