Iterates of the multidimensional Cesaro operator

The iterates of the Cesaro operator were studied recently (see Fontes, F. G. and Soils, F. J., Iterating the Cesaro operators, Proc. Amer. Math. Soc., 136 (2008), No. 6, 2147-2153) on some subsets of s(C), on (C[0, 1], C) and on C([0, infinity[, C). They proved that under suitable conditions the sequence of iterates converges to a constant function. In [Andras, Sz. and Rus, I. A., Iterates of Cesaro operators, Fixed Point Theory 11 (2010), No. 2, 171-178] the authors gave some more general results regarding the convergence of the iterates by proving that the Cesaro operator is a contraction on a dense subset of (C[0, 1],B), equipped with a well chosen norm, where B is a Banach space. The convergence of iterates for some general averaging operators involving one variable functions was also investigated by Sz. Andras and I. A. Rus in [Iterates of Cesaro operators, Fixed Point Theory 11 (2010), No. 2, 171-178]. The aim of this paper is to prove similar results involving Cesaro operators and general averaging operators for several variable functions. The proofs are suggested by the characterization theorem of weakly Picard operators on an L-space (see Rus, I. A., Picard operators and applications, Sci. Math. Jpn., 58 (2003), 191-219) and the method can be applied also in the study of some singular integral equations.