Deviation bounds for additive functionals of Markov processes

Patrick Cattiaux; Arnaud Guillin

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Jachymski showed that the set $\{(x, y) \in 𝐜_{0} \times 𝐜_{0} : {(\sum_{i = 1}^{n} α (i) x (i) y (i))}_{n = 1}^{\infty} is bounded\}$ is either a meager subset of $𝐜_{0} \times 𝐜_{0}$ or is equal to $𝐜_{0} \times 𝐜_{0}$ . In the paper we generalize this result by considering more general spaces than $𝐜_{0}$ , namely $𝐂_{0} (X)$ , the space of all continuous functions which vanish at infinity, and $𝐂_{b} (X)$ , the space of all continuous bounded functions. Moreover, we replace the meagerness by $σ$ -porosity.

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Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type

Silvia I. Hartzstein, Beatriz E. Viviani (2002)

Commentationes Mathematicae Universitatis Carolinae

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In the setting of spaces of homogeneous-type, we define the Integral, $I_{φ}$ , and Derivative, $D_{φ}$ , operators of order $φ$ , where $φ$ is a function of positive lower type and upper type less than $1$ , and show that $I_{φ}$ and $D_{φ}$ are bounded from Lipschitz spaces $Λ^{ξ}$ to $Λ^{ξ φ}$ and $Λ^{ξ / φ}$ respectively, with suitable restrictions on the quasi-increasing function $ξ$ in each case. We also prove that $I_{φ}$ and $D_{φ}$ are bounded from the generalized Besov ${\dot{B}}_{p}^{ψ, q}$ , with $1 \leq p, q < \infty$ , and Triebel-Lizorkin spaces ${\dot{F}}_{p}^{ψ, q}$ , with $1 < p, q < \infty$ , of order $ψ$ to those of order...