Krein condition and the Hilbert transform

高温下的经典 beta 系综及相关特征值过程与 Markov–Krein 变换

Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform

本文利用 Markov–Krein 变换刻画经典 beta 系综及其特征值过程在高温下的极限，识别出三类系综对应的极限测度及测度值过程。

马尔可夫-克莱因变换在经典卷积下的稳定性

Stability of the Markov-Krein Transform Under Classical Convolutions

该研究证明广义马尔可夫-克莱因变换在有限对数矩概率测度下的像对经典加法卷积和梅林卷积封闭，解决了加法高温卷积的正性猜想，其证明依赖通用预星形函数的卷积定理。

The structure of entrance laws for time-inhomogeneous Ornstein-Uhlenbeck Processes with Lévy Noise in Hilbert spaces

This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein-Uhlenbeck processes with Lévy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalising corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then prove that an arbitrary entrance law with finite weak first moments can be uniquely represented as an integral over extremals. It is proved that this can be derived from Dynkin's seminal work "Sufficient…

On the $l^p$-norm of the discrete Hilbert transform

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.

The dyadic and the continuous Hilbert transforms with values in Banach spaces

We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.

Discrete Hilbert Transform a la Gundy-Varopoulos

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on sto…