An inequality for a class of Markov processes

Let $\alpha \in (0,2)$ and consider the operator $\text{sL}$ given by $$\text{sL}f(x)=\int [f(x+h)-f(x)-1_{(\mathrm{\mid }h\mathrm{\mid }\le 1)}h\cdot \text{grad}f(x)]\frac{n(x,h)}{\mathrm{\mid }h{\mathrm{\mid }}^{d+\alpha }}\text{d}h,$$ where the term $1_{(|h|\leq 1)}h\cdot \grad f(x)$isnotpresentwhen$\alpha \in (0,1)$. Under some suitable assumptions on the kernel $n(x,h)$, we prove a Krylov-type inequality for processes associated with $\text{sL}$. As an application of the inequality, we prove the existence of a solution to the martingale problem for $\text{sL}$ without assuming any continuity of $n(x,h)$.