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Bao–Wulan multiplier problem for QK spaces

For the Möbius-invariant space QKQ_K on the unit disc, determine whether the pointwise multiplier algebra satisfies M(QK)=H∞∩QKlog⁡\mathcal{M}(Q_K)=H^\infty\cap Q_K^{\log}, where M(QK)={g:gf∈QK for every f∈QK and f↦gf is bounded on QK}\mathcal{M}(Q_K)=\{g:g f\in Q_K\text{ for every }f\in Q_K\text{ and }f\mapsto gf\text{ is bounded on }Q_K\} and QKlog⁡Q_K^{\log} is the logarithmic QKQ_K-class specified in the Bao–Wulan formulation.

References

Primary source

arXiv

Additional references

function-spacesharmonic-analysis
Posted

Progress summary

Refreshed
Claimed solved

An August 2026 preprint claims to settle the multiplier problem in these function spaces, but the result has not been independently checked.

The Bao–Wulan problem concerns whether the pointwise multipliers of QKQ_K equal the conjectured class H∞∩QKlog⁡H^\infty\cap Q_K^{\log}. It was highlighted as an open problem in the Bao–Wulan survey of 2021.

Known results

  • Li and Wulan (2010) proved separate logarithmic sufficient and necessary conditions for boundedness of the Volterra operator TgT_g on QKQ_K, leaving a gap.

August 2026 claimed resolution

A preprint dated 20 August 2026 claims a complete multiplier characterization using capacity conditions, equivalently a semidefinite-program formulation, and claims exact necessary-and-sufficient criteria for boundedness and compactness of TgT_g. It therefore claims to settle the multiplier conjecture and close the Volterra-operator gap, but no independent verification, referee report, correction, withdrawal, or retraction was found.

Current status (as of August 2026): The multiplier characterization and the corresponding Volterra criteria are claimed in an unrefereed preprint, but remain unverified.

Sources

Solutions 0

No solutions have been posted yet.