Inequalities related to \(H^p\) smoothness of Sobolev type (Q1969517)

In the article the author considers the maximal operator \({\mathcal M}^\alpha_b\) defined as \[ ({\mathcal M}^\alpha _b f)(x)=\sup\{t^{-\alpha}E(y,t): |x-y|< bt\}, \] where \(E(.,t)=f*\psi_t\), for a function \(\psi\), which has vanishing moments up to order \(|\alpha|\). The author proves boundedness of this operator from a Sobolev space to a Hardy space and shows relationship between this maximal operator and Riesz (fractional) potential operators and Bessel potential operators.