A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some L p ( μ ) , 1 ⩽ p < ∞ , where μ is a σ-finite measure. We study the convergence in the L p ( μ ) -norm and the almost everywhere convergence of the bilinear operators A n ( f 1 , f 2 ) = ( 1 2 n + 1 ∑ i = − n n T i f 1 ( x ) ) ( 1 2 n + 1 ∑ i = − n n T i f 2 ( x ) ) for functions f 1 ∈ L p 1 ( μ ) and f 2 ∈ L p 2 ( μ ) , 1 ⩽ p , 1 < p 1 , p 2 < ∞ , 1 / p 1 + 1 / p 2 = 1 / p . It turns out to be that the convergence in L p ( μ ) 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 ( μ ) -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.