Convergence of convolution operators and weighted averages in L(P) spaces

It is well known that it is possible to have pointwise convergence in some L_p spaces and not in others along the Individual Ergodic Theorem. We show that the same behavior is possible for perturbed moving averages and convolution operators induced by approximate identities. Furthermore, we study weighted versions of moving averages and differentiation operators. We address the question of optimality for the classes of weights used to assure that these operators satisfy weak type inequalities. We examine similarities and differences in the behavior of these two classes of operators with respect to existence of optimal weights.