Tauberian theorems for Cesàro summable double integrals over $ℝ^{2}_{+}$
Volume 138 / 2000
Abstract
Given ⨍ ∈ $L^1_loc (ℝ^2_+)$, denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these Tauberian conditions to be replaced either by Schmidt type slow decrease (or increase) conditions, or by Landau type one-sided Tauberian conditions. Second, we consider complex-valued functions and give a two-sided Tauberian condition which is necessary and sufficient in order that (**) follow from (*). In particular, this condition is satisfied if s(u,v) is slowly oscillating, or if f(x,y) obeys Landau type two-sided Tauberian conditions. At the end, we extend these results to the mixed case, where the (C, 1, 0) mean, that is, the average value of s(w,v) with respect to the first variable over the interval [0,u], is considered instead of $σ_11 (u,v) := σ(u,v)$
