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

in Analysis and PDE (in press)

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 ... [more ▼]

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. [less ▲]

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

in Stochastic Partial Differential Equations: Analysis and Computations (2023), 11(2), 685-713

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 ... [more ▼]

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. [less ▲]

Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian

E-print/Working paper (2022)

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

E-print/Working paper (2021)

We address some fundamental questions about geometric analysis on Riemannian manifolds. The L^p-Calderón-Zygmund inequality is one of the cornerstones in the regularity theory of elliptic equations, and ... [more ▼]

We address some fundamental questions about geometric analysis on Riemannian manifolds. The L^p-Calderón-Zygmund inequality is one of the cornerstones in the regularity theory of elliptic equations, and it has been asked under which geometric conditions it holds for a reasonable class of non-compact Riemannian manifolds, and to what extent assumptions on the derivative of curvature and on the injectivity radius of the manifold are necessary. In the present paper, for 1<p<2, we give a positive answer for the validity of the L^p-Calderón-Zygmund inequality on a Riemannian manifold assuming only a lower bound on the Ricci curvature. It is well known that this alone is not sufficient for p>2. In this case we complement the study of Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-Zygmund inequality under additional Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Bounds in the Kato class are integral conditions and much weaker than pointwise bounds. Throughout the proofs, probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role. [less ▲]