On Banach Spaces and Fréchet Spaces of Laplace–Stieltjes Integrals

We investigate the spaces of Laplace–Stieltjes integrals I(σ)=∫0∞f(x)exσdF(x), σ ∈ ℝ, F is a nonnegative nondecreasing unbounded function right continuous on [0, +∞), and f is a real-valued function on [0, +∞). This integral is a generalization of the Dirichlet series D(σ)=∑n=1∞dneλnσ with nonnegative exponents λn increasing to +∞ if F(x)=n(x)=∑λn≤x1, and f (x) = dn for x = λn and f (x) = 0 for x ≠ λn. For a positive continuous function h on [0, +∞) that increases to +∞, by LSh we denote a class of integrals I such that |f(x)| exp {xh(x)} → 0 as x → + ∞ and define ‖I‖h = sup {|f(x)| exp {xh(x)} : x ≥ 0}. We prove that if F ∈ V and ln F(x) = o(x) as x → +∞, then (LSh, ‖⋅‖h) is a nonuniformly convex Banach space. Some other properties of the space LSh and its dual space are also studied. As a consequence, we obtain results for the Banach spaces of Laplace–Stieltjes integrals of finite generalized order. Some results are refined in the case where I (σ) = D(σ). In addition, for fixed ϱ < +∞, we assume that S¯ ϱ is a class of entire Dirichlet series D (σ) such that their generalized order ϱα,β[D]≔limsupσ→+∞α(lnM(σD))β(σ)≤ϱ, where M(σD)=∑n=1∞|dn|eσλn and the functions α and β are positive, continuous on [x0, +∞), and increasing to +∞. Further, for q ∈ ℕ, let‖D‖ϱ;q=∑n=1∞|dn|exp{λnβ−1(α(λn)ϱ+1/q)},d(D1D2)=∑q=1∞12q‖D1−D2‖ϱ;q1+‖D1−D2‖ϱ;q. The space with the metric d is denoted by S¯ ϱ,d is a Fréchet space under certain conditions imposed on the functions α and β and the sequence (λn). © 2023, Springer Nature Switzerland AG.