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In this paper, we consider the following type of non-local  (pseudo-differential) operators $\LL $ on $\R^d$:  $$  \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x)\frac{\partial}{\partial x_j}) +  \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}}  (u(y)-u(x)) J(x, y) dy,  $$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$satisfying certain conditions. Corresponding to $\LL$ is a symmetric strongMarkov process $X$ on $\R^d$ that has both the diffusion component and purejump component. We establish a priori H\"older estimate for bounded parabolicfunctions of $\LL$ and parabolic Harnack principle for positive parabolicfunctions of $\LL$. Moreover, two-sided sharp heat kernel estimates are derivedfor such operator $\LL$ and jump-diffusion $X$. In particular, our resultsapply to the mixture of symmetric diffusion of uniformly elliptic divergenceform operator and mixed stable-like processes on $\R^d$. To establish theseresults, we employ methods from both probability theory and analysis.

A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps