Weighted Ergodic Theory And Dimension Free Estimates

Let T = {Ts}s∈R be a strongly continuous one-parameter group of positive operators acting on L(μ), where μ is a σ-finite measure and 1 ≤ p < ∞. We develop a weighted ergodic theory associated with T which parallels the classical theory of Ap-weights. A weighted transference principle is obtained in this setting and used to establish dimension free weighted inequalities for certain classical operators, including the higher order Riesz transforms.