Fractional Poisson equations and Ergodic theorems for fractional coboundaries

Yves Derriennic, Michael Lin

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53 Scopus citations

Abstract

For a given contraction T in a Banach space X and 0 < α < 1, we define the contraction Tα = Σj=1 ajTj where {aj} are the coefficients in the power series expansion (1 - t)α = 1 - Σj=1 ajtj in the open unit disk, which satisfy aj > 0 and Σj=1 = 1. The operator calculus justifies the notation (I - T)α := I - Tα (e.g., (I - T1/2)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 Tα. The fractional Poisson equation for T is the Poisson equation for Tα. 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 Σk=1 Tky/k1-α converges in norm, and conclude that limn ∥(1/n1-α) Σnk=1 Tky∥ = 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 Σk=1 Tk f/k converges a.e. For 1 < p < ∞, we prove that if ∈ 6 (I -T)α Lp with α > 1 - 1/p = 1/9, then Σk=1 Tk f/k1/p converges a.e., and thus (1/n1/p) Σnk=1 Tk f converges a.e. to zero. When f ∈ (I - T)1/q Lp (the case α = 1/q), we prove that (1/n1/p(log n)1/q) Σnk=1 Tk f converges a.e. to zero.

Original languageEnglish
Pages (from-to)93-130
Number of pages38
JournalIsrael Journal of Mathematics
Volume123
DOIs
StatePublished - 1 Jan 2001

ASJC Scopus subject areas

  • Mathematics (all)

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