Аннотация:
Consider parabolic pseudo-differential operators $L =\partial_t-p(x,D_x)$ of variable order
$\alpha(x)\leq 2$. The function $\alpha(x)$ is assumed to be smooth, but the symbol $p(x,\xi)$ is
not always differentiable with respect to $x.$ We will show the uniqueness of Markov
processes with the generator $L.$ The essential point in our study is to obtain the
$L^p$-estimate for resolvent operators associated with solutions to the martingale problem
for $L.$ We will show that, by making use of the theory of pseudo-differential
operators and a generalized Calderon–Zygmund inequality for singular integrals. As
a consequence of our study, the Markov process with the generator $L$ is constructed
and characterized. The Markov process may be called a stable-like process with
perturbation.