We provide quantitative weighted weak type estimates for non-integral square functions in the critical case p=2 in terms of the Aₚ and reverse Hölder constants associated to the weight. The method of proof uses a decoupling of the role of the weights via a quantitative version of Gehring's lemma. The results can be extended to other p in the range of boundedness of the square function at hand.
Mena et al. (Tue,) studied this question.