Let
$$d\geqslant 1$$
and
$$\alpha \in (0, 2)$$
. Consider the following non-local and non-symmetric Lévy-type operator on
$${\mathbb R}^d$$
:
$$\begin{aligned} {\fancyscript{L}}^\kappa _{\alpha }f(x):=\hbox {p.v.}\int _{{\mathbb R}^d}(f(x+z)-f(x)) \frac{\kappa (x,z)}{ |z|^{d+\alpha }} {\mathord {\mathrm{d}}}z, \end{aligned}$$
where
$$0<\kappa _0\leqslant \kappa (x,z)\leqslant \kappa _1, \kappa (x,z)=\kappa (x,-z)$$
, and
$$|\kappa (x,z)-\kappa (y,z)|\leqslant \kappa _2|x-y|^\beta $$
for some
$$\beta \in (0,1)$$
. Using Levi’s method, we construct the fundamental solution (also called heat kernel)
$$p^\kappa _\alpha (t, x, y)$$
of
$${\fancyscript{L}}^\kappa _\alpha $$
, and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that
$$p^\kappa _\alpha (t, x, y)$$
is jointly Hölder continuous in
$$(t, x)$$
. The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of
$${\fancyscript{L}}^\kappa _{\alpha }$$
gives rise a Feller process
$$\{X, {\mathbb P}_x, x\in {\mathbb R}^d\}$$
on
$${\mathbb R}^d$$
. We determine the Lévy system of
$$X$$
and show that
$${\mathbb P}_x$$
solves the martingale problem for
$$({\fancyscript{L}}^\kappa _{\alpha }, C^2_b({\mathbb R}^d))$$
. Furthermore, we show that the
$$C_0$$
-semigroup associated with
$${\fancyscript{L}}^\kappa _\alpha $$
is analytic in
$$L^p ({\mathbb R}^d)$$
for every
$$p\in [1,\infty )$$
. A maximum principle for solutions of the parabolic equation
$$\partial _t u ={\fancyscript{L}}^\kappa _\alpha u$$
is also established. As an application of the main result of this paper, sharp two-sided estimates for the transition density of the solution of
$${\mathord {\mathrm{d}}}X_t = A(X_{t-}) {\mathord {\mathrm{d}}}Y_t$$
is derived, where
$$Y$$
is a (rotationally) symmetric stable process on
$${\mathbb R}^d$$
and
$$A(x)$$
is a Hölder continuous
$$d\times d$$
matrix-valued function on
$${\mathbb R}^d$$
that is uniformly elliptic and bounded.