A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some L(p)(mu), 1 <= p < infinity, where mu is a sigma-finite measure. We study the convergence in the L(p)(mu)-norm and the almost everywhere convergence of the bilinear operators A(n)(f(1), f(2)) = (1/2n+1 Sigma(n)(i=-n) T(i)f(2)(x)) for functions f(1) is an element of L(p1)(mu) and f(2) is an element of L(p2)(mu), 1 <= p, 1 < p(1), p(2) < infinity, 1/p(1) + 1/p(2) = 1/p. It turns out to be that the convergence in L(p)(mu) is equivalent to the dominated estimate for the ergodic maximal operator associated to A(n) and to the uniform boundedness of the operators A(n). It is also shown that the convergence in the L(p)(mu)-norm implies the almost everywhere convergence. On the one hand, the key facts to prove these results are transference arguments and the connection with a new class of weights recently introduced by Lerner et al. (2009) [4]. On the other hand, our main result can be viewed as the ergodic counterpart of one of the main results in the above cited paper. (C) 2010 Elsevier Inc. All rights reserved.