Chiral Scale and Conformal Invariance in 2D QFT

Question

I am reading a paper by Hofman and Strominger. In the appendix A, I have reproduced the equations (A10). Now they made a statement that

"The Jacobi identity can be used to show that $O_h$ and $O_p$ are local operators with no explicit coordinate dependence."

I am not able to prove this statement. I proceeded as follows:

I considered the Jacobi identities $ i[H,i[D,h_{\pm}]]+\text{cyclic combinations}=0$, and $ i[\bar{P},i[D,h_{\pm}]]+\text{cyclic combinations}=0$.

But some terms appear like: $i[D,\partial_{\pm}h_{\pm}]$. But I don't know how to compute this commutator.

Can anyone help me to prove the statement?

This post imported from StackExchange Physics at 2015-05-18 21:04 (UTC), posted by SE-user layman