Weighted composition operators on the logarithmic Bloch spaces with iterated weights (Q1039545)

Let \(\ell(x)=(\log x)+1\) and \(\ell_n(x)\) donote the \(n\)-th iterate of \(\ell\). Here, \(\ell_0(x)=x\). Moreover, let \[ L_n(z)=\ell_{n-1}\left(\log\sqrt{\frac{1+|z|}{1-|z|}}+1\right). \] The logarithmic Bloch space \(\mathfrak{LB}^n\) with iterated weights is the set of all holomorphic functions in the open unit disk for which \(\sup_{z\in \mathbb D}(1-|z|^2) \prod_{k=1}^n L_k(z) |f '(z)|<\infty\). The authors characterize the boundedness and compactness of weighted composition operators on these spaces.