Heat kernels for time-dependent non-symmetric stable-like operators

Abstract

When studying non-symmetric nonlocal operators $$\mathcal{L}f(x)={\int }_{{\mathbf{R}}^d}\left(f(x+z)-f(x)-\nabla f(x)\cdot z{1}_{\{|z|\le 1\}}\right)\frac{\kappa (x,z)}{|z{|}^{d+\alpha }}dz,$$ where $0<\alpha <2$ and $\kappa (x,z)$ is a function on ${\mathbf{R}}^d\times {\mathbf{R}}^d$ that is bounded between two positive constants, it is customary to assume that $\kappa (x,z)$ is symmetric in $z$. In this paper, we study heat kernel of $\mathcal{L}$ and derive its two-sided sharp bounds without the symmetric assumption $\kappa (x,z)=\kappa (x,-z)$. In fact, we allow the kernel $\kappa$ to be time-dependent and also derive gradient estimate when $\beta \in (0\vee (1-\alpha ),1)$ as well as fractional derivative estimate of order $\theta \in (0,(\alpha +\beta )\wedge 2)$ for the heat kernel, where $\beta$ is the Hölder index of $x\mapsto \kappa (x,z)$. Moreover, when $\alpha \in (1,2)$, the drift perturbation with drift in Kato's class is also considered. As an application, when $\kappa (x,z)=\kappa (z)$ does not depend on $x$, we show the boundedness of nonlocal Riesz's transorfmation: for any $p>2d/(d+2\alpha )$, $$\Vert {\mathcal{L}}^{1/2}f{\Vert }_p\asymp \Vert \Gamma (f{)}^{1/2}{\Vert }_p,$$ where $\Gamma (f):=\frac{1}{2}\mathcal{L}(f^2)-f\mathcal{L}f$ is the carré du champ operator associated with $\mathcal{L}$, and ${\mathcal{L}}^{1/2}$ is the square root operator of $\mathcal{L}$ defined by using Bochner's subordination. Here $\asymp$ means that both sides are comparable up to a constant multiple.

References in corpus (1)

• Heat kernel estimates for non-symmetric stable-like processes

Cited by in corpus (1)

• Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel