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

When studying non-symmetric nonlocal operators $${\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{\kappa (x, z)}{|z|^{d+\alpha}} d z ,$$ where $0<\alpha<2$ and $\kappa (x, z)$ is a function on ${\bf R}^d\times {\bf R}^d$ 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 ${\cal L}$ 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 H\"older 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 $p>2d/(d+2\alpha)$, $$\| {\cal L}^{1/2}f\|_p\asymp \|\Gamma(f)^{1/2}\|_p,$$ where $\Gamma(f):=\frac{1}{2}{\cal L} (f^2)-f {\cal L} f$ is the carr\'e du champ operator associated with ${\cal L}$, and ${\cal L}^{1/2}$ is the square root operator of ${\cal L}$ defined by using Bochner's subordination. Here $\asymp$ means that both sides are comparable up to a constant multiple.