Fractional Poisson equations and ergodic theorems for fractional coboundaries |
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Authors: | Yves Derriennic Michael Lin |
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Institution: | (1) Université de Bretagne Occidentale, Brest, France;(2) Ben-Gurion University of the Negev, Beer-Sheva, Israel |
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Abstract: | For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT
α=Σ
j=1
∞
a
j
T
j
, where {a
j
} are the coefficients in the power series expansion (1-t)α=1-Σ
j=1
∞
a
j
t
j
in the open unit disk, which satisfya
j
>0 anda
j
>0 and Σ
j=1
∞
a
j
=1. The operator calculus justifies the notation(I−T)
α
:=I−T
α
(e.g., (I−T
1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for
T if there is anx∈X such that(I−T)
α
x=y, i.e.,y is a coboundary forT
α
. The fractional Poisson equation forT is the Poisson equation forT
α
. We show that if(I−T)X is not closed, then(I−T)
α
X strictly contains(I−T)X (but has the same closure).
ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ
k=1
∞
T
k
y/k
1-α converges in norm, and conclude that lim
n
‖(1/n
1-α)Σ
k=1
n
T
k
y‖=0 for suchy.
For a Dunford-Schwartz operatorT onL
1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T)
α
L
1 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 iff∈(I−T)
α
L
p
with α>1−1/p=1/q, then Σ
k=1
∞
T
k
f/k
1/p
converges a.e., and thus (1/n
1/p
) Σ
k=1
n
T
k
f converges a.e. to zero. Whenf∈(I−T)
1/q
L
p
(the case α=1/q), we prove that (1/n
1/p
(logn)1/q
)Σ
k=1
n
T
k
f converges a.e. to zero. |
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Keywords: | |
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