In this paper, we will consider composition operators defined between distinct Bergman spaces over planar domains. The smoothness on boundary of the domains plays an important role in our study. On one hand, an essential extension of Littlewood's Subordination Principle is obtained. Precisely, for each holomorphic map that is defined between bounded domains of smooth boundaries, the associated composition operator is always bounded. This essentially depends on a "standard decomposition" of holomorphic functions over a classical domain, bounded by finitely many disjoint circles. On the other hand, the situation becomes complex if domains with cusp boundary points are concerned, and there exists a link between the boundary behavior of the function symbol and the boundedness of the associated composition operator, where a detailed discussion is presented. Finally, we give estimates of norms for some classes of such composition operators. A deep interplay of function theory, geometry, and operator theory is revealed.

This work is partially supported by NSFC (12071134, 11471113). We thank the referee for his/her suggestions to improve the paper.