Iterates of the multidimensional Cesaro operator

The iterates of the Cesaro operator were studied recently (see Fontes, F. G. and Sol ` ´ıs, F. J., Iterating the Ces´aro 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, ∞[, 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 Ces`aro 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 Ces`aro 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