On the Estimates of the Density of the Purely Discontinuous Girsanov Transforms of $alpha$-Stable-like Processes

In this paper, we study the purely discontinuous Girsanov transforms which were discussed in Chen and Song cite{CS2} and Song cite{S3}. We show that the transition density of any purely discontinuous Girsanov transform of a $alpha$-stable-like process, which can be nonsymmetric, is comparable to the transition density of the $alpha$-stable-like process.