Full characterization of continuity of the integration operator in the unit disk

Characterize continuity of the integration operator \(J\) on weighted analytic spaces on the unit disk in terms of the growth of the quantity \(- (1-t)v'(t)/v(t)\), beyond the sufficient conditions currently available.

Background

In the unit-disk setting, the paper studies the differentiation and integration operators on spaces defined by the weight vM,Dv_{\mathbf{M},\mathbb{D}}. It proves continuity of the integration operator for the associated weight-sequence class by showing that the relevant differential quotient tends to infinity.

The discussion compares these results with prior characterizations for general disk weights. It explicitly notes that the existing results provide only sufficiency, leaving unresolved a complete necessary-and-sufficient characterization of continuity in terms of the differential growth expression.

References

Moreover, by Lemma 3.15, Cor. 3.16, Rem. 3.17 a full characterization of the continuity of $J$ in terms of the growth of the expression in crucialdiskcondition is unclear and only the sufficiency is shown.

crucialdiskcondition:

−(1−t)vM,D′(t)vM,D(t)=−(1−t)(exp⁡(−ωM,D(t)))′exp⁡(−ωM,D(t)).-\frac{(1-t)v'_{\mathbf{M},\mathbb{D}}(t)}{v_{\mathbf{M},\mathbb{D}}(t)}=-\frac{(1-t)(\exp(-\omega_{\mathbf{M},\mathbb{D}}(t)))'}{\exp(-\omega_{\mathbf{M},\mathbb{D}}(t))}.

— The differentiation and integration operator on weighted spaces of analytic functions in the weight sequence setting  (2610.11902 - Schindl, 8 Oct 2026) in Note following Theorem \ref{Thm30&31}, Section \ref{unitdiscsection}