arXiv subjects

Anton Thalmaier

Publications and source records attributed to Anton Thalmaier.

At least 19 recordsLinked to original sources

arXiv2026

$L^p$-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(Δ_μ^{(k)}+σ)^{-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 $σ$. In particular, in the unweighted case, the result confirms a conjecture of Baumgarth, Devyver and Güneysu~\cite{BDG-23}. As an application, we obtain Calderón--Zygmund inequalities for $p>2$ on weighted manifolds, which extends the recent work \cite{CCT} on manifolds without weight.

math.DG↗

arXiv2023

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 $\fracλ{n}\,\|ϕ\|_{\infty} \leq \|{\rm Hess}\ ϕ\|_{\infty}\leq c(D)λ\,\|ϕ\|_{\infty}$ holds for any Dirichlet eigenfunction $ϕ$ of $-Δ$ with eigenvalue $λ$. Our results provide the sharp Hessian estimate $\|{\rm Hess}\ ϕ\|_{\infty}\lesssim λ^{\frac{n+3}{4}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.

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 Weitzenböck curvature endomorphism. As an application, the Calderón-Zygmund inequality for $1< p\leq 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.

math.DG↗

arXiv2022

Variations of the sub-Riemannian distance on Sasakian manifolds with applications to coupling

On Sasakian manifolds with their naturally occurring sub-Riemannian structure, we consider parallel and mirror maps along geodesics of a taming Riemannian metric. We show that these transport maps have well-defined limits outside the sub-Riemannian cut-locus. Such maps are not related to parallel transport with respect to any connection. We use this map to obtain bounds on the second derivative of the sub-Riemannian distance. As an application, we get some preliminary result on couplings of sub-Riemannian Brownian motions.

math.DG↗

arXiv2022

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 $μ(d x)={\rm e}^{-V(x)} \mbox{vol}(d x)$ is a probability measure on $M$. Taking $μ$ 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↗

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 $L=Δ+Z$ 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 $L^2(μ)$ for some probability measure $μ$, 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↗

arXiv2022

Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds

We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the $L^p$-Calderón-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 Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-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

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↗

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 Schrödinger 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 integrability and exit times of diffusions on sub-Riemannian and metric measure spaces

In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider sub-Riemannian limits of Riemannian foliations. Secondly, we consider the non-smooth setting of $\mathrm{RCD}^*(K,N)$ spaces. In each case the necessary ingredients are an Itô formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrödinger operators.

math.PR↗

arXiv2020

Radial processes for sub-Riemannian Brownian motions and applications

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng's type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.

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↗

arXiv2020

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 Poincaré inequalities.

math.DG↗

arXiv2018

Gradient Estimates on Dirichlet Eigenfunctions

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\_1(D)$ and $c\_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c\_1(D)\sqrtλ\|ϕ\|\_\infty \le \|\nabla ϕ\|\_\infty\le c\_2(D)\sqrtλ \|ϕ\|\_\infty$ holds for any Dirichlet eigenfunction $ϕ$ of $-Δ$ with eigenvalue $λ$. In particular, when $D$ is convex with nonnegative Ricci curvature, this estimate holds for $c\_1(D)=\frac{1}{de}$ and $c\_2(D)=\sqrt{e}\left(\frac{\sqrt{2}}{\sqrtπ}+\frac{\sqrtπ}{4\sqrt{2}}\right)$. Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

math.PR↗

arXiv2018

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.

math.DG↗

arXiv2018

Derivative and divergence formulae for diffusion semigroups

For a semigroup $P_t$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_t(V(f))$, not involving derivatives of $f$, where $V$ is a vector field on $M$. For non-symmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.

math.PR↗

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↗

arXiv2018

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 $Δu$. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings.

math.AP↗