A sharp inequality for the square function
Let f *, S(f), and Mf, respectively, denote the dyadic maximal, square, and Hardy-Littlewood maximal functions of f. For Q a dyadic cube and V a non-negative function in \(L\) \(1_{loc}({\mathbb{R}}^ d\)), define \(Y(Q,V)=\int_{Q}(M(\chi_ QV)/\int_{Q}V)\int_{Q}V>0\), \(Y(Q,V)=1,\int_{Q}V=0\). Let \(\psi\) :[0,\(\infty)\mapsto [1,\infty)\) be increasing, with \(\psi (2x)\leq C_ 1\psi (x)\) for all x. Define \[ M_{\psi}V(x)=\sup_{Q\ni x,Qdyadic}\psi (\log Y(Q,V))\frac{1}{| Q|}\int_{Q}V. \] We show that if \(\sum^{\infty}_{k=0}\frac{1}{\psi (k)}\leq 1\) then (*) \(\int | f\quad *|^ 2Vdx\leq C(C_ 1,d)\int S^ 2(f)M_{\psi}Vdx\) for all \(f\in C_ 0^{\infty}({\mathbb{R}}^ d\)) and weights V. We show that (*) fails, for any \(C(C_ 1,d)<\infty\), if \(\sum 1/\psi (k)=\infty\). We obtain as a corollary a strong sufficient condition for the two-weight inequality \(\int | f\quad *|^ 2Vdx\leq \int S^ 2(f)Wdx\) to hold.
dyadic maximal function. Hardy-Littlewood maximal functions
scientific article; zbMATH DE number 4042516