Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces (Q969286)

The Marcinkiewicz integral with variable kernel is defined by \[ \mu_{\Omega}(f)(x) = \Big( \int_{0}^{\infty} \Big| \int_{ | x-y | \leq t} \frac{\Omega (x,x-y)}{| x-y |^{n-1}}f(y)dy \Big|^2 \frac{dt}{t^3} \Big)^{1/2}. \] A function \(\Omega (x,z)\) defined on \({\mathbb R}^n \times {\mathbb R}^n\) is said to be in \(L^{\infty}({\mathbb R}^n) \times L^q({\mathbb S}^{n-1}) \), if \(\Omega\) satisfies the following three conditions: \[ \begin{aligned} &\Omega (x, \lambda z) = \Omega (x,z) \quad \text{for any}\quad \lambda >0; \\ &\| \Omega \|_{ L^{\infty}({\mathbb R}^n) \times L^q({\mathbb S}^{n1-}) } = \sup_{r \geq 0, y \in {\mathbb R}^n} \Big( \int_{{\mathbb S}^{n-1}} | \Omega( rz' +y, z' )|^q d\sigma(z') \Big)^{1/q} < \infty ; \\ &\int_{{\mathbb S}^{n-1}}\Omega (x,z') d\sigma(z')=0 \quad \text{for any}\quad x \in {\mathbb R}^n. \end{aligned} \] \textit{Y. Ding, C.-C. Lin} and \textit{S. Shao} [Indiana Univ. Math. J. 53, No. 3, 805--821 (2004; Zbl 1074.42004)] proved \(L^2\) boundedness of \(\mu_{\Omega}\). The authors consider the boundedness on Hardy spaces. For \(0 < \alpha \leq 1\), a function \(\Omega\) is called to satisfy \(L^{1,\alpha}\)-\textit{Dini condition} if \[ \int_0^1 \frac{\omega(t)}{t^{1+\alpha}}dt < \infty, \] where \[ \omega(t)= \sup_{r >0, y \in {\mathbb R}^n, | O | < t} \int_{{\mathbb S}^{n-1}} | \Omega (rz'+y, Oz') - \Omega(rz' +y, z')| d\sigma(z'), \] and \(O\) is a rotation in \({\mathbb R}^n\) with \(| O | = \| O - I \|\), where \(I\) is the identity operator. They prove the following. Let \(\Omega \in L^{\infty}({\mathbb R}^n) \times L^q({\mathbb S}^{n-1}) \) with \(q> 2(n-1)/n\), and let \(\Omega\) satisfy \(L^{1, \alpha}\)-Dini condition. Then \(\mu_{\Omega}\) is bounded from Hardy space \(H^p({\mathbb R}^n)\) to \(L^p({\mathbb R}^n)\) if \( \max \{ 2n/(2n+1), n/(n + \alpha) \} < p <1\).