On Banach Spaces and Fréchet Spaces of Laplace–Stieltjes Integrals

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2023-02, Vol.270 (2), p.280-293
• Main Authors: Kuryliak, A. O., Sheremeta, M. M.
• Format: Article
• Language: English
• Subjects: Banach spaces; Continuity (mathematics); Dirichlet problem; Integrals; Mathematical functions; Mathematics; Mathematics and Statistics
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-023-06346-9

We investigate the spaces of Laplace–Stieltjes integrals I σ = ∫ 0 ∞ f x e xσ dF 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 ∞ d n e λ n σ with nonnegative exponents λ n increasing to +∞ if F x = n x = ∑ λ n ≤ x 1 , and f ( x ) = d n for x = λ n and f ( x ) = 0 for x ≠ λ n . For a positive continuous function h on [0, +∞) that increases to +∞, by LS h 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 ( LS h , ‖⋅‖ h ) is a nonuniformly convex Banach space. Some other properties of the space LS h 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 ϱ α , β D ≔ lim sup σ → + ∞ α ln M σ D β σ ≤ ϱ , where M σ D = ∑ n = 1 ∞ d n e σ λ n and the functions α and β are positive, continuous on [ x 0 , +∞), and increasing to +∞. Further, for q ∈ ℕ, let D ϱ ; q = ∑ n = 1 ∞ d n exp λ n β − 1 α λ n ϱ + 1 / q , d D 1 D 2 = ∑ q = 1 ∞ 1 2 q D 1 − D 2 ϱ ; q 1 + D 1 − D 2 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 (λ n ).