On commutators of Marcinkiewicz integrals with rough kernel
Let \(h(t)\) be a bounded function on \(\mathbb{R}^+\) and \(b\) a function in the \(\text{BMO}(\mathbb{R}^n)\) space. In this paper, the authors consider the higher-order commutator \(\mu^m_{\Omega,b}(f)\) of the Marcinkiewicz integral defined by \[ \mu^m_{\Omega,b}(f)(x)= \Biggl\{\int^\infty_0|F^m_{t, b}(x)|^2 t^{-3} dt\Biggr\}^{1/2}, \] where \[ F^m_{t,b}(x)= \int_{|x-y|< t} h(|x-y|) \Omega(x-y)|x-y|^{-n+1} \{b(x)- b(y)\}^m f(y) dy \] and \(\Omega\) is an \(L^q\), \(q> 1\), function on the sphere \(S^{n-1}\) that satisfies the following conditions: \[ \Omega(tx)= \Omega(x),\quad\text{for any }t\in \mathbb{R}^+;\;\int_{S^{n-1}}\Omega(x) d\sigma(x)= 0. \] The authors also consider its related operators \(\mu^{*,m}_{\Omega, \lambda,b}\) that is corresponding to the Stein \(g^*_\lambda\)-function and \(\mu^m_{\Omega, S,b}\) that is corresponding to the Lusin area integral. The authors establish various boundedness properties of these operators on the weighted spaces \(L^p(\mathbb{R}^n,\omega)\), where the weigh \(\omega\) depends on \(p\) and \(q\).
scientific article; zbMATH DE number 1866670
scientific article