Laplace Transforms for Analytic Functions in Tubular Domains

Assume that 0 < p < ∞ and that B is a connected nonempty open set in ℝ n , and that A p ( B ) is the vector space of all holomorphic functions F in the tubular domains ℝ n + i B such that for any compact set K ⊂ B , ‖ y ↦ ‖ x ↦ F ( x + i y ) ‖ L p ( ℝ n ) ‖ L ( K ) < ∞ , so A p ( B ) is a Fréchet space with the Heine-Borel property, its topology is induced by a complete invariant metric, is not locally bounded, and hence is not normal. Furthermore, if 1 ≤ p ≤ 2, then the element F of A p ( B ) can be written as a Laplace transform of some function f ∈ L (ℝ n ).