Linear dimension dependence for stabilization in Haar system Hardy spaces

Determine whether, for every Haar system Hardy space Y, there exists a constant C = C(Γ, δ, η, Y) > 0 such that for all n and all N ≥ C n, the conclusion of Proposition 4.1 (Stabilization of Haar multipliers) holds; namely, for every linear operator T: Y_N → Y_N with ||T|| ≤ Γ, there exists a scalar c with |c| ≤ ||T|| such that c I_{Y_n} projectionally factors through T with constant 1 and error 3 η δ, and if T has δ-large positive diagonal with respect to the Haar basis, then c ≥ (1 − η) δ. Equivalently, ascertain whether the factorization conclusions of Theorem 1.1 can be obtained under a linear (in n) bound on N for all Haar system Hardy spaces.

Background

The paper proves quantitative factorization results for operators on finite-dimensional subspaces Y_n spanned by the first n+1 levels of the Haar system within Haar system Hardy spaces. The main theorem provides quasi-polynomial dimension dependence in the general case (N grows like n^2, up to parameters) and polynomial (linear) dependence when the Haar system is unconditional.

Proposition 4.1 (stabilization) shows that after diagonalization, one can stabilize a Haar multiplier D to a scalar multiple of the identity on Y_n, at the cost of a dimension bound N that grows quadratically in n in general. The authors improve this to linear dependence in the unconditional case, and also note an L^1-specific improvement by a different argument.

The open question asks whether this improvement to linear dependence can be obtained for every Haar system Hardy space, i.e., whether the quadratic dependence in the general stabilization step can be reduced to a linear dependence, which would in turn yield linear bounds in Theorem 1.1 for all Haar system Hardy spaces.