Weak A2 sufficiency conjecture for two-sided operator-orbit Poisson dextrodual expansions

Establish that for any finite absolutely continuous Borel measure μ on [0,1) whose Radon–Nikodym derivative g satisfies the weak A2-type inequality (for every interval I ⊂ [0,1)) (1/|I|)∫_I g(x) dx · (1/|I|)∫_I (1/g(x)) χ_{ {x : g(x) > 0} } dx ≤ C, there exist an invertible bounded operator T on L^2(μ) and a function g0 ∈ L^2(μ) such that the bi-infinite operator orbit {T^n g0}_{n∈ℤ} is Poisson dextrodual to the classical exponentials {e^{2π i n x}}_{n∈ℤ} in L^2(μ).

Background

After deriving necessary conditions (including T = M_{e^{2π i x}}, absolute continuity of μ with Radon–Nikodym derivative g, and a weak A2-type condition), the authors conjecture that this weak condition is also sufficient to guarantee the existence of two-sided operator-orbit systems that are Poisson dextrodual to the exponentials.

The conjecture aims to bridge the gap between necessary conditions and known sufficiency results that typically require the full Muckenhoupt A2 condition or stronger assumptions.