Triple product formula and the subconvexity bound of Triple product L-function in level aspect

In this paper we derive a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalize Venkatesh and Woodbury's work on the subconvexity bound of triple product $L$-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters, and general special values of local epsilon factors.