For weighted Bergman spaces A_alpha^p, determine a necessary-and-sufficient characterization of the zero-free functions f∈ A_alpha^p whose reciprocal 1/f also belongs to A_alpha^p. Equivalently, determine the reciprocal pairs f,1/f in A_alpha^p across the relevant parameter ranges.

Reciprocal problem on weighted Bergman spaces

For weighted Bergman spaces $A_{\alpha}^{p}$ , determine a necessary-and-sufficient characterization of the zero-free functions $f\in A_{\alpha}^{p}$ whose reciprocal $1/f$ also belongs to $A_{\alpha}^{p}$ . Equivalently, determine the reciprocal pairs $f,1/f$ in $A_{\alpha}^{p}$ across the relevant parameter ranges.

References

Primary source

arXiv

Additional references

The Reciprocal Problem on Weighted Bergman Spaces — arXiv — Guangfu Cao, Li He, Shuqing Zhang

complex-analysis function-spaces

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper settles several important cases of the problem, but the full weighted Bergman-space question remains open.

The problem asks for a reciprocal characterization in weighted Bergman spaces. Guangfu Cao, Li He, and Shuqing Zhang report progress across several parameter regimes without claiming a complete solution.

Recent partial solutions

The paper identifies ranges handled by analytic Besov composition methods, solves the $d=3$ Drury–Arveson case, and characterizes the $d=4$ case. These are claimed advances, not a resolution for all parameters.

Current status (as of September 2026): Several parameter regimes, including the $d=3$ Drury–Arveson case and the $d=4$ case, are claimed to be settled, while the full weighted Bergman-space problem remains open.

Sources

No solutions have been posted yet.