The martingale problem for geometric stable-like processes

We prove that the martingale problem is well posed for pure-jump Lévy-type operators of the form $$(\mathcal{L}f)(x)={\int }_{{\mathbb{R}}^d\setminus \{0\}}(f(x+h)-f(x)-(\mathrm{\nabla }f(x)\cdot h)1_{\mathrm{\parallel }h\mathrm{\parallel }<1})K(x,h)dh,$$ where $K(x,\cdot )$ is a jump kernel of the form $K(x,h)\sim \frac{l(\mathrm{\parallel }h\mathrm{\parallel })}{\mathrm{\parallel }h{\mathrm{\parallel }}^d}$ for each $x\in {\mathbb{R}}^d,\mathrm{\parallel }h\mathrm{\parallel }<1$, and $l$ is a positive function that is slowly varying at $0$, under suitable assumptions on $K$. This includes jump kernels such as those of $\alpha$-geometric stable processes, $\alpha \in (0,2]$.