We study the compactness of composition operators on Bergman spaces certain bounded convex domains in $\mathbb{C}^n$ with non-trivial analytic discs contained boundary. As a consequence we characterize that operator continuous symbol (up to closure) space polydisc.

In this paper we prove a $K$-homology index theorem for the Toeplitz operators obtained from multishifts of Bergman space on several classes egg-like domains. This generalizes our with Douglas and Yu unit ball.

The inclusions between the Besov spaces $B^q$, Bloch space $\mathcal{B}$ and standard weighted Bergman $A^p_\alpha$ are completely understood, but norms of corresponding inclusion operators in general unknown. In this work, we compute or estimate asymptotically these inclusions.