Joint Carleson measure and the difference of composition operators on A_α^p(B_n)

We introduce a concept of joint Carleson measure and characterize when the difference of two composition operators on A_α^p(B_n), the weighted Bergman space over the unit ball B_n in Cⁿ, is bounded or compact. We apply this joint Carleson measure characterization to composition operators with smooth symbols and construct an interesting example which shows that the boundedness or the compactness depends on p when n ≥ 2. This is in sharp contrast with the single composition operator case where the boundedness or the compactness is independent of p>0. Moreover, the compact difference on the weighted Bergman spaces over the unit disc is known to be independent of p>0, and the compact difference on A_α^p(B_n) is known to be independent of p>0 if each composition operator is bounded on A_β^p(Bⁿ) for some -1< β < α [2].