Reverse inequality between c_{b,1} and c_{b,2} in general de Branges–Rovnyak spaces

Establish whether, for an arbitrary de Branges–Rovnyak space H(b) with \|b\|_{H^\infty}=1 and any set E ⊂ 𝕋, the reverse inequality c_{b,2}(E) ≤ c_{b,1}(E) holds, where c_{b,1}(E) = inf{ \|f\|^2_{H(b)} : f ∈ H(b), |f| ≥ 1 almost everywhere on a neighborhood of E } and c_{b,2}(E) = inf{ \|f\|^2_{H(b)} : f ∈ H(b), |f| = 1 almost everywhere on a neighborhood of E }.

Background

The capacities c_{b,1} and c_{b,2} (introduced by Fricain–Grivaux) are tools for studying cyclicity in de Branges–Rovnyak spaces. Trivially, c_{b,1}(E) ≤ c_{b,2}(E) for all E. However, the general validity of the reverse inequality has remained unclear.

In this paper, the authors prove comparability of c_{b,1} and c_{b,2} when H(b) coincides with a harmonically weighted Dirichlet space D_μ (Proposition 7.1), providing a partial answer to a question of Fricain–Grivaux about comparability in every H(b) space. The full reverse inequality in general H(b) remains open.