Carleson measure characterizations on analytic QK(p; q) spaces

In this paper, we study composition operators on Bloch space and QK(p, q) spaces. We give a Carleson measure characterization on QK(p, q) spaces, then we use this Carleson measure characterization of the compact compositions on QK(p, q) spaces to show that every compact composition operator on QK(p, q) spaces is compact on Bloch space. Moreover, necessary and sufficient condition for C? from the Bloch space B to a general class of analytic functions QK(p, q) to be compact is given in terms of the map ?.