Extension of weak-A2 weights to A2 weights

Ascertain whether any nonnegative weight g on [0,1) that 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 admits an extension to a Muckenhoupt A2 weight on [0,1) that agrees with g almost everywhere on its positive set.

Background

To establish sufficiency for Poisson dextrodual operator-orbit expansions, the authors consider a weaker inequality than the Muckenhoupt A2 condition and discuss the possibility of extending such weights to true A2 weights.

They note that while satisfying the weak inequality is strictly weaker than the A2 condition, it is not clear whether such weights admit A2 extensions; they consequently invoke results requiring a stronger assumption (g^{1+ε} satisfying the inequality) to obtain partial sufficiency.