The Reciprocal Problem on Weighted Bergman Spaces

@misc{cao2026reciprocal,
  title = {{The Reciprocal Problem on Weighted Bergman Spaces}},
  author = {Guangfu Cao and Li He and Shuqing Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19896},
  eprint = {2609.19896},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19896},
}

TY  - UNPB
TI  - The Reciprocal Problem on Weighted Bergman Spaces
AU  - Guangfu Cao
AU  - Li He
AU  - Shuqing Zhang
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.19896
UR  - https://arxiv.org/abs/2609.19896
AB  - We study the reciprocal problem for weighted Bergman spaces: if f∈ A_α^p(Bn) and inf_Bn|f|>0, does it follow that 1/f∈ A_α^p(Bn)? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in A_α^p(D) have the reciprocal property for all α∈ R and p≥ 1, where D denotes the unit disk in C. Moreover, functions in A_α^p(Bn) have the reciprocal property for all α∈ R when n=2 and p=2. In addition, we resolve the reciprocal problem in the three-dimensional Drury--Arveson space H_3^2 and obtain an equivalent condition in the four-dimensional space H_4^2. We also settle the reciprocal problem for general A_α^p(Bn) under some additional conditions.
ER  -