Ergodic theorem in variable Lebesgue spaces
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Ergodic theorem in variable Lebesgue spaces Przemysław Górka1
© The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract We prove an Ergodic Theorem in variable exponent Lebesgue spaces, whenever the exponent is invariant under the transformation. Moreover, a counterexample is provided which shows that the norm convergence fails to hold for an arbitrary exponent. Keywords measure
Lebesgue spaces with variable exponents · Ergodic theorem · Probability
AMS Subject Classification
28D05 · 54C35
1 Introduction Variable exponent Lebesgue and Sobolev spaces are natural extensions of classical constant exponent L p -spaces. Such kind of theory finds many applications for example in nonlinear elastic mechanics [23], electrorheological fluids [20] or image restoration [18]. During the last decade Lebesgue and Sobolev spaces with variable exponents have been intensively studied; see for instance the surveys [5,21]. In particular, the Sobolev inequalities have been shown for variable exponent spaces on Euclidean spaces (see [4,7] and [15]) and on Riemannian manifolds (see [9] and [11]). Recently, the theory of variable exponent spaces has been extended to metric measure spaces, see e.g. [8,10,16,17,19]. Moreover, the theory of Lebesgue spaces with variable exponent on probability spaces exists as well, see e.g. [1] In this article we investigate Birkhoff’s Ergodic Theorem in the context of variable Lebesgue spaces. Let us mention that Ergodic theorems in spaces other than Lebesgue spaces have been studied in the past (see e.g. [2,13,14,22]). We organize this paper as follows. In the next section we review some definitions and present the theory of variable exponent spaces. In the third section we present and prove the main result.
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Przemysław Górka [email protected] Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland
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P. Górka
2 Variable exponent Lebesgue spaces In this section we recall some basic facts and notation about variable exponent Lebesgue spaces. Most of the properties of these spaces can be found in the book of Cruz-Uribe and Fiorenza [3] and in the book of Diening et al. [6]. Let (, μ) be a σ -finite, complete measure space. By a variable exponent we shall mean a bounded measurable function p : → [1, ∞). We put p + = ess sup p(x),
p − = ess inf p(x). x∈
x∈
The variable exponent Lebesgue space L p(·) () consists of those μ-measurable functions f : → R for which semimodular | f (x)| p(x) dμ(x) ρ p(·) ( f ) =
is finite. This is a Banach space with respect to the following Luxemburg norm f f p(·) = inf λ > 0 : ρ p(·) ≤1 , λ where f ∈ L p(·) (). Variable Lebesgue space is a special case of the Musielak–Orlicz spaces. When the variable exponent p is constant, then L p(·) () is an ordinary Lebesgue space. It is needed very often to pass between norm and semimodular. An important property of the variable Lebesgue spaces is the so-called ball property: f L p(·)
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