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Heat Kernel for Fractional Diffusion Operators with Perturbations

Let $L$ be an elliptic differential operator on a complete connected Riemannian manifold $M$ such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let $L^{(å)}$ be the $å$-stable subordination of $L$ for $å\in (1,2).$ We found some classes $\mathbb K_å^{\gg,\bb} (\bb,\gg\in [0,å))$ of time-space functions containing the Kato class, such that for any measurable $b: [0,\infty)\times M\to TM$ and $c: [0,\infty)\times M\to M$ with $|b|, c\in \mathbb K_å^{1,1},$ the operator $$L_{b,c}^{(å)}(t,x):= L^{(å)}(x)+ <b(t,x),\nn \cdot> +c(t,x),\ \ (t,x)\in [0,\infty)\times M$$ has a unique heat kernel $p_{b,c}^{(å)}(t,x;s,y), 0\le s<t, x,y\in M$, which is jointly continuous and satisfies &\ff{t-s}{C\{\rr(x,y)\lor (t-s)^{\frac{1}å}\}^{d+å}}\le p_{b,c}^{(å)}(t,x;s,y)\le \ff{C(t-s)}{{\rr(x,y)\lor (t-s)^{\frac{1}å}}^{d+å}}, & \big|\nn_x p_{b,c}^{(å)}(t,x; s,y)\big|\le \ff{C(t-s)^{\ff{å-1}å}}{{\rr(x,y)\lor (t-s)^{\frac{1}å}}^{d+å}}, 0\le s<t,\ x,y\in M for some constant $C>1$, where $\rr$ is the Riemannian distance. The estimate of $\nabla_yp^{(å)}_{b,c}$ and the Hölder continuity of $\nn_x p_{b,c}^{(å)}$ are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where $L=\DD$ on $\R^d$ and $b,c$ are time-independent.