Boundedness characterization of Volterra-type operators on area Nevanlinna spaces

For each $1\le p<\infty$ and $\alpha>-1$ , determine exactly which analytic symbols $g$ make the Volterra-type operators $J_g$ and $I_g$ metrically bounded on the area Nevanlinna space $N_\alpha^p$ . In the standard notation $J_g f(z)=\int_0^z f(\zeta)g'(\zeta)\,d\zeta$ and $I_g f(z)=\int_0^z f'(\zeta)g(\zeta)\,d\zeta$ , the claimed characterization is $J_g$ is metrically bounded on $N_\alpha^p\iff g\in\mathcal B$ and $I_g$ is metrically bounded on $N_\alpha^p\iff g\in H^\infty$ , where $\mathcal B$ is the Bloch space.

References

Primary source

arXiv

Additional references

Metrically bounded Volterra-type operators on area Nevanlinna spaces — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed preprint claims to complete the boundedness characterization, but the result has not been independently verified.

The problem asks for exact symbol conditions characterizing boundedness of both Volterra-type operators on area Nevanlinna spaces. Earlier work gave only partial characterizations; a new preprint claims to settle the remaining cases.

August 2026 claimed completion

The preprint Metrically bounded Volterra-type operators on area Nevanlinna spaces claims exact symbol classes for both operators, thereby completing the characterization problem. It is presented as an unrefereed preprint, so the claimed solution is unverified.

Current status (as of August 2026): An unrefereed preprint claims a complete solution, but the exact characterization remains unverified.

Sources

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