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The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for…
In this paper, we construct and analyze Bessel and Flett potentials associated with the heat and Poisson semigroups in the framework of the $(k,1)$-generalized Fourier transform. We establish fundamental properties of these potentials and…
We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the…
In this paper, we establish a parabolic Harnack inequality for positive solutions of the $\phi$-heat equation and prove Gaussian upper and lower bounds for the $\phi$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci…
We first study the Lipschitz spaces $\Lambda_{d}^\beta$ associated with the Dunkl metric, $\beta\in(0,1)$, and prove that it is a proper subspace of the classical Lipschitz spaces $\Lambda^\beta$ on $\mathbb R^N$, as the Dunkl metric and…
We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and…
In this thesis we describe a type of metric space called an Euclidean polyhedral complex. We define a Dirichlet form on it; this is used to give a corresponding heat kernel. We provide a uniform small time Poincare inequality for complexes…
This paper introduces heat semigroups of topological Markov chains and Cuntz-Krieger algebras by means of spectral noncommutative geometry. Using recent advances on the logarithmic Dirichlet Laplacian on Ahlfors regular metric-measure…
The description of the phase space of relativistic particles coupled to three-dimensional Einstein gravity requires momenta which are coordinates on a group manifold rather than on ordinary Minkowski space. The corresponding field theory…
In this paper, we study the asymptotic behavior of the heat kernel with respect to the Witten Laplacian. We introduce the localization and the scaling technique in semi-classical analysis, and study the semi-classical asymptotic behavior of…
We construct an index of first-order, self-adjoint, elliptic differential operators in the $K$-theory of a Fr\'echet algebra of smooth kernels with faster than exponential off-diagonal decay. We show that this index can be represented by an…
The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and…
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We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic…
We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the…
Motivated by the SYZ picture for the collapsing of elliptic K3 surfaces, we study heat kernels under semi-flat collapse of Ricci-flat K\"ahler manifolds (X_t, g(t)) fibered by flat 2-tori over a surface B with a finite discriminant set D.…
The aim of this work is the study of magnetic trajectories on nilmanifolds. The magnetic equation is written and the corresponding solutions are found for a family of invariant Lorentz forces on a 2-step nilpotent Lie group equipped with a…
Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of K\"ahler manifolds, we prove…
In this paper, we prove Hamilton's Harnack inequality and the gradient estimates of the logarithmic heat kernel for the Witten Laplacian on complete Riemainnian manifolds. As applications, we prove the $W$-entropy formula for the Witten…
Asymptotic expansions of heat kernels and heat traces of Schr\"odinger operators on non-compact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very…