\(H^p_w- L^p_w\) boundedness of Marcinkiewicz integral (Q2455078)

The Marcinkiewicz integral is essentially a Littlewood-Paley \(g\)-function, which plays a very important role in harmonic analysis. In this paper, the authors give weaker smoothness conditions, so-called the \(L^q\)-Dini condition of order \(\alpha\), on \(\Omega\) to imply the \(H^p_{\omega}-L^p_{\omega}\) boundedness of Marcinkiewicz integral operators \(\mu_{\Omega}\), where \(\omega\) belongs to the Muckenhoupt weight class.