Bulletin of the
Korean Mathematical Society BKMS

A holomorphic function $F$ defined on the unit disc belongs to $\mathcal A^{p,\alpha}(0 < p < \alpha< ~\infty$. For boundedness of the composition operator defined by $C_f g =g\circ f$ mapping Blochs into $\mathcal A^{p,\alpha}$, the following (1) is a sufficient condition while (2) is a necessary condition. $$\int_0^1 \frac 1{1-r} \big(1+\log \frac 1{1-r}\big) ^{- \alpha}~M_p( r , \lambda \circ f)^p ~ dr ~< ~ \infty,\tag{1}$$ $$ \int_0^1 \frac 1{1-r} \big(1+\log \frac 1{1-r}\big)^{- \alpha +p} (1-r)^p~ M_p \left(r, f^\sharp \right)^p dr ~< ~\infty.\tag{2}$$

Keywords: composition operator, Bloch space, weighted Bergman space

MSC numbers: 32A37