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.