Complete the Green-kernel criterion

Determine whether the family of power-weight Green-kernel estimates can be replaced by a single beta-independent Schur weight, whether sharpness can be recovered from rowwise non-annihilation without column limits or a dominating envelope, and which hypotheses in the Green-kernel criteria are necessary.

Background

The Green-kernel existence theorems give sufficient conditions for transparency, absorption, and sharpness of the regularity index. One theorem controls individual inverse-kernel columns, while a stronger formulation uses weighted row sums and allows cancellation among entries.

The authors explicitly leave open the possibility of a unified Schur estimate, a rowwise replacement for column-limit non-annihilation, removal of the envelope assumption, and a necessity theory for the stated hypotheses.

References

Three questions remain.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Open Problem ‘Boundary of the Green criterion’, Section 5.4

The equality $\Gamma_G=\mathcal N_V$ is not settled by the absorption argument.

— Weighted potential and Green systems with Dini coefficients and primitive growth  (2610.26656 - Hu et al., 19 Oct 2026) in Remark \ref{r7:green-open-boundary}, Section \ref{r7:green-section}