Iteration of composition operators on small Bergman spaces of Dirichlet series

The Hilbert spaces $\mathscr{H}_{w}$ consisiting of Dirichlet series $F(s)=\sum_{ n = 1}^\infty a_n n^{ -s }$ that satisfty $\sum_{ n=1 }^\infty | a_n |^2/ w_n < \infty$, with $\{w_n\}_n$ of average order $\log_j n$ (the $j$-fold logarithm of $n$), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon--Hedenmalm theorem on such $\mathscr{H}_w$ from an iterative point of view. By that theorem, the composition operators are generated by functions of the form $\Phi(s) = c_0s + \phi(s)$, where $c_0$ is a nonnegative integer and $\phi$ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when $c_0=0$. It is verified for every integer $j\geqslant 1$, real $\alpha>0$ and $\{w_n\}_{n}$ having average order $(\log_j^+ n)^\alpha$ , that the composition operators map $\mathscr{H}_w$ into a scale of $\mathscr{H}_{w'}$ with $w_n'$ having average order $( \log_{j+1}^+n)^\alpha$. The case $j=1$ can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.