\(L^ p\) weighted norm inequalities for the square function, \(0<p<2\)
The author proves the following theorem: For every \(0<p<2\) there is a \(C(p,d)<\infty\), such that \(\int | f^*|^ pVdx\leq C(p,d)\int S^ p(f)MVdx\) for all \(f\in {\mathcal C}_ 0^{\infty}\) and non-negative \(V\in L^ 1_{loc}({\mathbb{R}}^ d)\), where \(Q\subset {\mathbb{R}}^ d\) is a dyadic cube with sidelength \(\ell (Q)\) and Lebesgue measure \(| Q|\), \(f\in L^ 1_{loc}({\mathbb{R}}^ d)\), \(f_ Q=(1/| Q|)\int_{Q}f,\) \(f_ k=\sum_{\ell (Q)=2^{-k}}f_ Q\chi_ Q,\) k integer, \(\chi_ Q\) is the characteristic function of Q, \(f^*(x)=\sup_{k}| f_ k(x)|\), \(a_ Q(f)=(f_{k+1}-f_ k)\cdot \chi_ Q\), \(S(f)=(\sum_{Q\ni x}\| a_ Q(f)\|^ 2_ 2/| Q|)^{1/2}\) is the dyadic square function and \(Mf(x)=\sup_{Q\ni x}| f|_ Q\) the Hardy-Littlewood maximal operator.
scientific article; zbMATH DE number 4120742