On the martingale problem for pseudo-differential operators of variable order

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.