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Register a new userOn local convexity in $mathbb{L}^0$ and switching probability measures
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In the paper, we investigate the following fundamental question. For a set $mathcal{K}$ in $mathbb{L}^0(mathbb{P})$, when does there exist an equivalent probability measure $mathbb{Q}$ such that $mathcal{K}$ is uniformly integrable in $mathbb{L}^1(mathbb{Q})$. Specifically, let $mathcal{K}$ be a convex bounded positive set in $mathbb{L}^1(mathbb{P})$. Kardaras [6] asked the following two questions: (1) If the relative $mathbb{L}^0(mathbb{P})$-topology is locally convex on $mathcal{K}$, does there exist $mathbb{Q}sim mathbb{P}$ such that the $mathbb{L}^0(mathbb{Q})$- and $mathbb{L}^1(mathbb{Q})$-topologies agree on ${mathcal{K}}$? (2) If $mathcal{K}$ is closed in the $mathbb{L}^0(mathbb{P})$-topology and there exists $mathbb{Q}sim mathbb{P}$ such that the $mathbb{L}^0(mathbb{Q})$- and $mathbb{L}^1(mathbb{Q})$-topologies agree on $mathcal{K}$, does there exist $mathbb{Q}sim mathbb{P}$ such that $mathcal{K}$ is $mathbb{Q}$-uniformly integrable? In the paper, we show that, no matter $mathcal{K}$ is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of $mathbb{Q}simmathbb{P}$ satisfying these desired properties. We also investigate the peculiar effects of $mathcal{K}$ being positive.
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We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.
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The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{e} constant. First we revisit E. Milmans result [40] on the link between weak (Poincar{e} or concentration) inequalities and Cheegers inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{e} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the logconcave case, to reduce to the case of the Poincar{e} inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...).
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good and bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
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