Marcinkiewicz integral on weighted Hardy spaces
Let \(\Omega\) be a homogeneous function of degree zero on \(\mathbb R^n\) where \(n \geq 2\). We assume that \(\Omega \in L^1(S^{n-1})\) and \(\int_{S^{n-1}} \Omega(x')dx' =0\) where \(x' = x/| x |\). Then the Marcinkiewicz integral operator \(\mu_{\Omega}\) on \(\mathbb R^d\) is defined by \[ \mu_{\Omega} f(x) = \Bigl( \int_0^{\infty}| F_{\Omega, t}(f,x) |^2 \frac{dt}{t^3} \Bigr)^{1/2}, \] where \[ F_{\Omega, t}(f,x) = \int_{| x-y | \leq t} \frac{\Omega (x-y)}{| x-y |^{n-1}} f(y)dy. \] Several weighted \(L^p\) estimates for \(\mu_{\Omega}\) are known. \textit{A. Torchinsky} and \textit{S. Wang} [Colloq. Math. 60/61, No. 1, 235--243 (1990; Zbl 0731.42019)] showed that if \(\Omega \in \text{Lip}_{\alpha} \;(0 < \alpha \leq 1)\) and \(w \in A_p\), then \(\mu_{\Omega}\) is bounded on \(L^p_w\) for \(1<p< \infty\). \textit{Y. Ding}, \textit{D. Fan} and \textit{Y. Pan} [Indiana Univ. Math. J. 48, No. 3, 1037--1055 (1999; Zbl 0949.42014)] showed that if \(\Omega \in L^q(S^{n-1})\) and \(w^{q'} \in A_p\) where \(1<q \leq \infty\) and \(1/q + 1/q' =1\), then \(\mu_{\Omega}\) is bounded on \(L^p_w\) for \(1<p< \infty\). The authors obtain the following: If \(\Omega\) satisfies an \(L^q\)-Dini condition \((q>1)\) and \(w^{q'} \in A_1\), then \(\mu_{\Omega}\) is a bounded operator from \(H^1_w\) to \(L^1_w\). They also consider the \(H^p_w \rightarrow L^p_w\) boundedness when \(\Omega \in \text{Lip}_{\alpha}\).
scientific article; zbMATH DE number 2004962