Uniqueness for the martingale problem associated with pure jump processes of variable order

Let $L$ be the operator defined on $C^2$ functions by $$Lf(x)=\int [f(x+h)-f(x)-1_{(\mathrm{\mid }h\mathrm{\mid }\le 1)}\mathrm{\nabla }f(x)\cdot h]\frac{n(x,h)}{\mathrm{\mid }h{\mathrm{\mid }}^{d+\alpha (x)}}dh\mathrm{.}$$ This is an operator of variable order and the corresponding process is of pure jump type. We consider the martingale problem associated with $L$. Sufficient conditions for existence and uniqueness are given. Transition density estimates for $\alpha$-stable processes are also obtained.