Related papers: Heat Kernel estimates for some elliptic operators …
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 prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator \begin{align*} \mathcal L&=\mbox{Tr }\left(AD^2\right)+\frac{\left(v,\nabla\right)}y, \end{align*} in the half-space…
In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) L\'evy processes on half spaces for all $t>0$. These L\'evy processes may…
In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps…
The goal of this paper is to establish sharp two-sided estimates on the heat kernels of two types of purely discontinuous symmetric Markov processes in the upper half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary. The…
In this paper, we study the geometry associated with Schroedinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both…
We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big)…
In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators $m-(m^{2/\alpha}-\Delta)^{\alpha/2}$] in $C^{1,1}$ open sets. Here $m>0$ and…
One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{\Theta, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with…
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The…
We consider the Dirichlet form given by \sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+&\int_{\bR^d\times \bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption…
We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace--Beltrami operator on Damek--Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually…
For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the…
We give estimates for positive solutions for the Schr\"odinger equation $(\Delta_\mu+W)u=0$ on a wide class of parabolic weighted manifolds $(M, d\mu)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined…
Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such that $\frac{Q}{Q+N} <…
We study reflected jump diffusions on Ahlfors regular domains in general metric measure spaces. Under the condition that the Dirichlet form on the ambient space satisfies a capacity upper bound estimate, we construct an extension operator…
The goal of this paper is to study the action of the heat operator on the Heisenberg group H^d, and in particular to characterize Besov spaces of negative index on H^d in terms of the heat kernel. That characterization can be extended to…
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…
The calculation of heat-kernel coefficients with the classical DeWitt algorithm has been discussed. We present the explicit form of the coefficients up to $h_5$ in the general case and up to $h_7^{min}$ for the minimal parts. The results…
We give a unified and optimized proof of the sharp bounds for the Jacobi heat kernel, which were obtained gradually in several papers in recent years. We lay particular emphasis on tracing and estimating all constants appearing throughout…