The converse compactness assertion for composition operators on analytic Campanato spaces

Let $\mathcal{CA}_p$ and $\mathcal{CA}_q$ be the analytic Campanato spaces appearing in Theorem

, and let $C_\phi$ denote the composition operator induced by an analytic self-map $\phi$ of $\mathbb D$. For $(p,q)\in [0,2)\times[0,2)$ outside the parameter range covered by the converse part of Theorem

, namely

$$[0,2)\times[0,2)\setminus\big([0,2)\times[1,1]\cup(1,2)\times[0,2)\big),$$

The converse compactness assertion. If the necessary condition (e1) and condition (e2) from Theorem

hold, then $C_\phi:\mathcal{CA}_p\mapsto\mathcal{CA}_q$ is compact. The assertion proposes that the converse part of Theorem

remains valid throughout this remaining parameter range. The paper presents this as an extension of the partial compactness result, while the precise conditions (e1) and (e2) are stated earlier in the source.