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Composition operators on small spaces. (English) Zbl 1114.47028
Let ${A}_{\alpha ,s}^{p}$ denote the space of holomorphic functions $f$ on the unit disk such that ${\int }_{𝔻}|{ℛ}^{s}{f\left(z\right)|}^{p}{\left(1-|z|}^{2}{\right)}^{\alpha }\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)<\infty$, that is to say that its $s$-fractional derivative ${ℛ}^{s}f\left(z\right)={\sum }_{n=0}^{\infty }{\left(n+1\right)}^{\alpha }{a}_{n}{z}^{n}$ belongs to ${A}_{\alpha }^{p}$. The authors analyze the boundedness and compactness of ${C}_{\phi }$ on the spaces ${A}_{\alpha ,s}^{p}$ in some particular situations. Motivated by the notion of “suitably small” Banach space, they denote by $ℛ$ the set of parameters $\left(s,p,\alpha \right)$ such that either $sp>\alpha +2$ or $sp=\alpha +2,\phantom{\rule{4pt}{0ex}}0. Their main results establish that if $\left(s-1,p,\alpha \right)\in ℛ$ (where $s$ is assumed to be a positive integer for $\alpha =-1$) and $\phi \in {A}_{\alpha ,s}^{p}$, then the composition operator ${C}_{\phi }$ is bounded on ${A}_{\alpha ,s-k}^{p}$ for all $k\in {ℤ}^{+},\phantom{\rule{4pt}{0ex}}k\le s$. In particular, they obtain in such a case that ${C}_{\phi }$ is bounded on ${A}_{\alpha ,s}^{p}$ if and only if $\phi \in {A}_{\alpha ,s}^{p}$. As a byproduct, they get that ${A}_{\alpha ,s}^{p}$ is an algebra under multiplication for $\left(s,p,\alpha \right)\in ℛ$. Among other interesting results in the paper, the authors also give a complete description of symbols for the boundedness of ${C}_{\phi }$ on the Zygmund class ${{\Lambda }}_{1}$ of functions such that $|{f}^{\text{'}\text{'}}\left(z\right)|=O\left(\frac{1}{1-|z|}\right)$. Finally, some examples concerning the boundedness of ${C}_{\phi }$ on ${A}_{\alpha ,s}^{p}$ if $\alpha +1\le sp\le \alpha +2+p$ are provided.
##### MSC:
 47B33 Composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions 30D55 H (sup p)-classes (MSC2000)
##### Keywords:
composition operators; holomorphic Sobolev spaces