Composition operators from logarithmic Bloch spaces to weighted Bloch spaces
We characterize the analytic self-maps $ϕ$ of the unit disk ${\Bbb D}$ in ${\Bbb C}$ that induce continuous composition operators $C_ϕ$ from the log-Bloch space $\mathcal{B}^{\log}({\Bbb D})$ to $μ$-Bloch spaces ${\mathcal B}^μ({\Bbb D})$ in terms of the sequence of quotients of the $μ$-Bloch semi-norm of the $n$th power of $ϕ$ and the log-Bloch semi-norm (norm) of the $n$th power $F_n$ of the identity function on ${\Bbb D}$, where $μ:{\Bbb D}\rightarrow (0,\infty)$ is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of $C_ϕ$ between these spaces, thus characterizing $ϕ$ such that $C_ϕ$ is compact. After finding a pairwise norm equivalent family of log-Bloch type spaces that are defined on the unit ball ${\Bbb B}_n$ of ${\Bbb C}^n$ and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for $C_ϕ$ acting on ${\mathcal B}^{\log}$ to the case when $C_ϕ$ acts on these more general log-Bloch-type spaces.