Orlicz integrability of additive functionals of Harris ergodic Markov chains

For a Harris ergodic Markov chain $(X_n)_{n\ge 0}$, on a general state space, started from the so called small measure or from the stationary distribution we provide optimal estimates for Orlicz norms of sums $\sum_{i=0}^\tau f(X_i)$, where $\tau$ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function $f$ (wrt the stationary distribution) and the regeneration time $\tau$ (wrt the small measure). We provide applications to tail estimates for additive functionals of the chain $(X_n)$ generated by unbounded functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen)

Vector-valued extensions of operators through multilinear limited range extrapolation

We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an $m$-(sub)linear operator $$T:L^{p_1}(w_1^{p_1})\times\cdots\times L^{p_m}(w_m^{p_m})\to L^p(w^p)$$ for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces $L^{p}(w^p;X)$ for a wide class of Banach function spaces $X$, which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.Comment: 21 pages. Minor modifications. To appear in Journal of Fourier Analysis and Application

On the monotone properties of general affine surface areas under the Steiner symmetrization

In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is monotone increasing, while the $L_{\psi}$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\phi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal\'{o} point) at the origin