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.
Highlights
Variable exponent Lebesgue and Sobolev spaces are natural extensions of classical constant exponent L p-spaces
During the last decade Lebesgue and Sobolev spaces with variable exponents have been intensively studied; see for instance the surveys [5,21]
The Sobolev inequalities have been shown for variable exponent spaces on Euclidean spaces and on Riemannian manifolds
Summary
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]. The theory of Lebesgue spaces with variable exponent on probability spaces exists as well, see e.g. In this article we investigate Birkhoff’s Ergodic Theorem in the context of variable Lebesgue spaces. 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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