Weighted inequalities for the dyadic square function without dyadic \(A_{\infty}\) (Q1100383)

Let \(f^*\) and \(S(f)\) denote, respectively, the dyadic maximal and square functions of f. We prove weighted norm inequalities of the form \[ \int | f\quad *|^ 2Vdx\leq C(\tilde M)\int S^ 2(f)\tilde MVdx, \] valid for all \(f\in C_ 0^{\infty}({\mathbb{R}}^ d\)) and non-negative \(V\in L\) \(1_{loc}({\mathbb{R}}^ d\)), where \(\tilde M\) is an appropriate ``maximal function''. In general, the \(\tilde MV\)'s we consider do not have the (dyadic) Muckenhoupt \(A_{\infty}\) property. In addition, we give a necessary condition and a similar-looking sufficient condition for the two-weight inequality \(\int | f^*|^ 2Vdx\leq C\int S^ 2(f)Wdx\) to hold.