Related papers: L p -estimates for the heat semigroup on different…

We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates…

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami…

We consider a self-adjoint non-negative operator $H$ in a Hilbert space $\mathsf{L}^2(X,{\rm d}\mu)$. We assume that the semigroup $(\mathrm{e}^{-t H})_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form…

On a doubling metric measure space endowed with a "carr\'e du champ", we consider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup and scale-invariant $L^p$ Poincar\'e inequalities $(P_p)$. We show that the combination of…

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…

We study solutions for the Hodge laplace equation $\Delta u=\omega $ on $p$ forms with $\displaystyle L^{r}$ estimates for $\displaystyle r>1.$ Our main hypothesis is that $\Delta $ has a spectral gap in $\displaystyle L^{2}.$ We use this…

Let $G=(V, E)$ be a locally finite connected graph satisfying curvature-dimension conditions ($CDE(n, 0)$ or its strengthened version $CDE'(n, 0))$) and polynomial volume growth conditions of degree $m$. We systematically establish sharp…

On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic one-forms, the Gaussian heat kernel upper…

Let $G = N \rtimes A$, where $N$ is a stratified group and $A = \mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian…

Let $\alpha>0$, $H=(-\triangle)^{\alpha}+V(x)$, $V(x)$ belongs to the higher order Kato class $K_{2\alpha}(\mathbbm{R}^n)$. For $1\leq p\leq \infty$, we prove a polynomial upper bound of $\|e^{-itH}(H+M)^{-\beta}\|_{L^p, L^p}$ in terms of…

Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let D be a bounded $C^{1,1}$ open set in $\mathbb{R}^d$ and b an $\mathbb{R}^d$-valued function on $\mathbb{R}^d$ whose components are in a certain Kato class of the rotationally symmetric…

Let $M$ be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of $L^p$-boundedness of the Riesz transform, $p\in (2,\infty)$. We also provide counter-examples…

This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the…

We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\le k\le 2n+1$, let $\Delta_k$ denote the Hodge Laplacian restricted to $k$-forms. Our first…

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to…

The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - \Delta\right) u \leq A u^p + B u $$ with small initial data in borderline Morrey norms over a…

For Riemannian symmetric spaces $X=G/K$ of noncompact type, we show that for all left $K$-invariant $f\in L^1(X)$, the functions $\|h_t\|_{L^p(X)}^{-1}(f\ast h_t-M_p(f)h_t)$ (with $h_t$ being the heat kernel of $X$) converges to zero in…

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of…

The $L^p$-boundedness for $p>2$ of the covariant Riesz transform on differential forms is proved for a class of non-compact weighted Riemannian manifolds under certain curvature and volume growth conditions, which in particular settles a…

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…