Tauberian theorems for Cesaro summable double integrals over R-+(2)

Given f is an element of L-loc(1)(R-+(2)), denote by s(w,z) its integral ov er the rectangle [0,w] x [0, z] and by sigma(u,v) its (C,1, 1) mean, that i s, the average value of s(w, z) over [0,u] x [0, v], where u, v, w, z > 0. Our permanent assumption is that (*) sigma(u, v) --> A as u, v --> infinity , where A is a finite number. First, we consider real-valued functions f and give one-sided Tauberian con ditions which are necessary and sufficient in order that the convergence (* *) s(u,v) --> A as u,v --> w follow from (*). Corollaries allow these Taube rian conditions to be replaced either by Schmidt type slow decrease (or inc rease) 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 oscil lating, or if f(z, 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 varia ble over the interval [0, u], is considered instead of sigma(11)(u, v) := s igma(u, v).