Tauberian conditions under which convergence of integrals follows from summability (C, 1) over R₊

Given ƒ ∈ L_loc¹(R₊), we define s(t):= ∫₀^tf(x)dx and σ (t):= ∫₀^ts(u)du for t>0 Our permanent assumption is that σ(t) → A as t → ∞, where A is a finite number

First, we consider real-valued functions, and prove that s(t) → A as t → ∞ if and only if two one-sided Tauberian conditions are satisfied. In particular, these two conditions are satisfied if s(t) is slowly decreasing (or increasing) in the sense of R. Schmidt; in particular, if ƒ(x) obeys a Landau type one-sided Tauberian condition.

Second, we extend these results for complex-valued functions by giving a two-sided Tauberian condition, being necessary and sufficient in order that σ(t) → A imply s(t) → A as t → ∞. In particular, this condition is satisfied if s(t) is slowly oscillating; in particular if ƒ(x) obeys a Landau type two-sided Tauberian condition