Reciprocal property in the unresolved weighted Bergman parameter range

Determine whether every holomorphic function f in the weighted Bergman space A_\alpha^p(\mathbb B_n) satisfying \inf_{z\in\mathbb B_n}|f(z)|>0 also has reciprocal 1/f\in A_\alpha^p(\mathbb B_n) for parameters -n-1<\alpha\leq-p-1.

Background

The paper studies whether a function in a weighted Bergman space whose modulus is bounded below has a reciprocal in the same space. It establishes the reciprocal property for \alpha>-p-1 and for \alpha<-n-1, and treats the endpoint \alpha=-n-1 under additional critical embedding assumptions. After these results, the authors identify the remaining range -n-1<\alpha\leq-p-1 as unresolved.

Resolving this range requires controlling the lower-order products that occur in the higher-order radial derivative formula for R^m(1/f). The authors explain that these cases are related to endpoint embedding problems for analytic Besov spaces and are not settled by the methods developed in the paper.