Fractional Poisson equations and Ergodic theorems for fractional coboundaries

For a given contraction T in a Banach space X and 0 < α < 1, we define the contraction T_α = Σ^∞_j=1 a_jT^j where {a_j} are the coefficients in the power series expansion (1 - t)^α = 1 - Σ^∞_j=1 a_jt^j in the open unit disk, which satisfy a_j > 0 and Σ^∞_j=1 = 1. The operator calculus justifies the notation (I - T)^α := I - T_α (e.g., (I - T_1/2)² = I - T). A vector y ∈ X is called an α-fractional coboundary for T if there is an x ∈ X such that (I -T)^αx = y, i.e., y is a coboundary for T_α. The fractional Poisson equation for T is the Poisson equation for T_α. We show that if (I - T)X is not closed, then (I - T)^αX strictly contains (I - T)X (but has the same closure). For T mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove that y ∈ X is an α-fractional coboundary if and only if Σ^∞_k=1 T^ky/k^1-α converges in norm, and conclude that lim_n ∥(1/n^1-α) Σⁿ_k=1 T^ky∥ = 0 for such y. For a Dunford-Schwartz operator T on L₁ of a probability space, we consider also a.e. convergence. We prove that if f ∈ (I - T)^α L₁ for some 0 < α < 1, then the one-sided Hilbert transform Σ^∞_k=1 T^k f/k converges a.e. For 1 < p < ∞, we prove that if ∈ 6 (I -T)^α L_p with α > 1 - 1/p = 1/9, then Σ^∞_k=1 T^k f/k^1/p converges a.e., and thus (1/n^1/p) Σⁿ_k=1 T^k f converges a.e. to zero. When f ∈ (I - T)^1/q L_p (the case α = 1/q), we prove that (1/n^1/p(log n)^1/q) Σⁿ_k=1 T^k f converges a.e. to zero.