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Boundedness of commutators for the Marcinkiewicz integrals on Hardy spaces. (English) Zbl 1005.42011

The authors consider the higher-order commutator \(\mu^m_{\Omega,b}\), \(m\in\mathbb{N}\), formed by the Marcinkiewicz integral \(\mu_\Omega\) and a BMO function \(b\), and given by \[ \mu^m_{\Omega, b}(f)(x)= \Biggl(\int^\infty_0 |F^m_{\Omega,b,t}(x)|^2 {dt\over t^3}\Biggr)^{1/2},\quad x\in\mathbb{R}^n, \] where \[ F^m_{\Omega,b,t}(x)= \int_{|x-y|\leq t} {\Omega(x- y)\over|x-y|^{n- 1}} (b(x)- b(y))^m f(y) dy. \] The main results of the paper establish conditions on the kernel \(\Omega\) under which, for certain range of \(p\) in \([0,1]\), the operator \(\mu^m_{\Omega,b}\) is bounded from a class of atomic-Hardy spaces \(H^p_{b^m}(\mathbb{R}^n)\) (that it is a subspace of the standard real Hardy space) to \(L^p(\mathbb{R}^n)\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces
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