It is well known that some important Markov semi-groups have a “regularization effect” – as for example th hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in L^p for p > 1. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in \(\mathbb {R}^{n}\), as well as for the \(M/M/\infty \) queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

众所周知，一些重要的马尔可夫半群具有“正则化效应”——例如布尔超立方体上的噪声算子的超收缩性或实线上的 Ornstein-Uhlenbeck 半群，它适用于L ^p为p> 1. Talagrand 在 1989 年猜想布尔超立方体上的噪声算子对于刚好可积的函数具有更微妙的正则化性质，但这个猜想仍然是开放的。尽管如此，Eldan-Lee 和 Lehec 近年来通过将 Ornstein-Uhlenbeck 半群的 log-Hessian 的不等式与下高斯测度。在这项工作中，我们探讨了这种现象有多普遍的问题。具体来说，我们的第一个目标是探索这些成分对\(\mathbb {R}^{n}\) 中的一些扩散半群以及\(M/M/\infty \)非负整数上的队列和正实数线上的 Laguerre 半群。我们的第二个目标是证明这些设置的一维正则化效果，即使在这些成分无效的情况下也是如此。