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Li-Juan Cheng

Publications and source records attributed to Li-Juan Cheng.

At least 19 recordsLinked to original sources
arXiv2026

Gradient Estimates for Neumann Semigroups on Manifolds with Boundary under Unbounded Curvature Conditions

This paper establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature $\mathrm{Ric}_Z \geq K$ and second fundamental form $\mathrm{II} \geq \sigma$ for potentially unbounded functions $K$ and $\sigma$. We then apply these formulas to derive pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds. Both convex and non-convex boundary cases are treated. In the non-convex case, the boundary contribution is controlled by a conformal change of metric and an exponential estimate for the boundary local time.

math.DG↗
arXiv2025

$L^p$-Boundedness of the Covariant Riesz Transform on Differential Forms for $p>2$

We establish the \(L^p\)-boundedness, for \(p>2\), of the covariant Riesz transform \(\nabla(\Delta_\mu^{(k)}+\sigma)^{-1/2} \) on differential forms over a class of complete weighted Riemannian manifolds. The proof is based on an heat-kernel criterion involving local volume doubling, heat kernel upper estimates, Kato-type curvature control, and gradient bounds for the heat semigroup on forms. Under curvature-dimension assumptions and Kato-type curvature bounds, this criterion applies and yields boundedness for all sufficiently large \(\sigma\). In particular, in the unweighted case, the result confirms a conjecture of Baumgarth, Devyver and G\"uneysu~\cite{BDG-23}. As an application, we obtain Calder\'on--Zygmund inequalities for \(p>2\) on weighted manifolds, which extends the recent work \cite{CCT} on manifolds without weight.

math.DG↗
arXiv2025

Hessian matrix estimates of heat-type equations via Bismut-Stroock Hessian formula

In this paper, we establish a new global Hessian matrix estimate for heat-type equations on Riemannian manifolds using a Bismut-type Hessian formula. Our results feature fully explicit coefficients as well as delay / growth rate functions. These estimates yield two key applications: a novel backward weak Harnack inequality and a precise pointwise Hessian estimate for eigenfunctions.

math.AP↗
arXiv2024

Probability Versions of Li-Yau Type Inequalities and Applications

By using stochastic analysis, two probability versions of Li-Yau type inequalities are established for diffusion semigroups on a manifold possibly with (non-convex) boundary. The inequalities are explicitly given by the Bakry-Emery curvature-dimension, as well as the lower bound of the second fundamental form if the boundary exists. As applications, a number of global and local estimates are presented, which extend or improve existing ones derived for manifolds without boundary. Compared with the maximum principle technique developed in the literature, the probabilistic argument we used is more straightforward and hence considerably simpler.

math.PR↗
arXiv2022

Covariant Riesz transform on differential forms for $1<p\leq2$

In this paper, we study $L^p$-boundedness ($1<p\leq 2$) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of $L^p$-boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenb\"{o}ck curvature endomorphism. As an application, the Calder\'on-Zygmund inequality for $1< p\leq 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.

math.DG↗
arXiv2022

Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq \|{\rm Hess}\ \phi\|_{\infty}\leq c(D)\lambda \,\|\phi\|_{\infty}$ holds for any Dirichlet eigenfunction $\phi$ of $-\Delta$ with eigenvalue $\lambda$. Our results provide the sharp Hessian estimate $\|{\rm Hess}\ \phi\|_{\infty}\lesssim \lambda^{\frac{n+3}{4}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.

math.PR↗
arXiv2022

Second Order Bismut formulae and applications to Neumann semigroups on manifolds

Let $M$ be a complete connected Riemannian manifold with boundary $\partial M$, and let $P_t$ be the Neumann semigroup generated by $\frac{ 1}{ 2} L$ where $L=\Delta+Z$ for a $C^1$-vector field $Z$ on $M$. We establish Bismut type formulae for $LP_t f$ and ${\rm Hess}_{P_tf}$ and present estimates of these quantities under suitable curvature conditions. In case when $P_t$ is symmetric in $L^2(\mu)$ for some probability measure $\mu$, a new type of log-Sobolev inequality is established which links the relative entropy $H$, the Stein discrepancy $S$, and relative Fisher information $I$, generalizing the authors' recent work in the case without boundary.

math.PR↗
arXiv2021

Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds.

math.PR↗
arXiv2021

Hessian heat kernel estimates and Calder\'{o}n-Zygmund inequalities on complete Riemannian manifolds

We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the $L^p$-Calder\'{o}n-Zygmund inequalities extend to a reasonable class of non-compact Riemannian manifolds without the assumption of a positive injectivity radius. In the present paper, we give a positive answer for $1 2$, we complement the study in G\"{u}neysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calder\'{o}n-Zygmund inequality by adding Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role throughout the proofs.

math.DG↗
arXiv2021

Some inequalities on Riemannian manifolds linking Entropy,Fisher information, Stein discrepancy and Wasserstein distance

For a complete connected Riemannian manifold $M$ let $V\in C^2(M)$ be such that $\mu(d x)={\rm e}^{-V(x)} \mbox{vol}(d x)$ is a probability measure on $M$. Taking $\mu$ as reference measure, we derive inequalities for probability measures on $M$ linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.

math.DG↗
arXiv2020

Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

Let $M$ be a differentiable manifold endowed with a family of complete Riemannian metrics $g(t)$ evolving under a geometric flow over the time interval $[0,T[$. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on $M$ which is generated by a Schr\"{o}dinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.

math.PR↗
arXiv2020

Exponential contraction in Wasserstein distance on static and evolving manifolds

In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not necessarily required to be non-negative. Compared to the results of Wang (2016), we focus on explicit estimates for the exponential contraction rate. Moreover, we show that our results extend to manifolds evolving under a geometric flow. As application, for the time-inhomogeneous semigroups, we obtain a gradient estimate with an exponential contraction rate under weak curvature conditions, as well as uniqueness of the corresponding evolution system of measures.

math.DG↗
arXiv2019

Functional inequalities on path space of sub-Riemannian manifolds and applications

For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a natural gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequalities on path space analogous to what has been established in Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev and Poincar\'e inequalities.

math.DG↗
arXiv2018

Uniform gradient estimates on manifolds with a boundary and applications

We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux's argument, and uniform quantitative gradient estimates, firstly for $C^2_b$ functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians.

math.FA↗
arXiv2017

Evolution systems of measures and semigroup properties on evolving manifolds

An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete Riemannian metrics, parametrized by $I=(-\infty,T)$. Given an additional $C^{1,1}$ family of vector fields $(Z_t)_{t\in I}$ on $M$. We study the family of operators $L_t=\Delta_t +Z_t $ where $\Delta_t$ denotes the Laplacian with respect to the metric $g_t$. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by $L_t$, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.

math.PR↗
arXiv2017

Quantitative $C^1$-estimates by Bismut formulae

For a $C^2$ function $u$ and an elliptic operator $L$, we prove a quantitative estimate for the derivative $du$ in terms of local bounds on $u$ and $Lu$. An integral version of this estimate is then used to derive a condition for the zero-mean value property of $\Delta u$. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings.

math.AP↗
arXiv2016

Characterization of pinched Ricci curvature by functional inequalities

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, $L^p$-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities.

math.PR↗