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Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak-Orlicz Hardy spaces. (English) Zbl 1405.42035

Summary: Let \(\varphi:\mathbb{R}^{n}\times[0,\infty)\to[0,\infty)\) satisfy that \(\varphi(x,\cdot)\), for any given \(x\in\mathbb{R}^{n}\), is an Orlicz function and that \(\varphi(\cdot ,t)\) is a Muckenhoupt \(A_{\infty}\) weight uniformly in \(t\in(0,\infty)\). The weak Musielak-Orlicz Hardy space \({WH}^{\varphi}(\mathbb{R}^{n})\) is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak-Orlicz space \({WL}^{\varphi}(\mathbb{R}^{n})\). For parameter \(\rho\in(0,\infty)\) and measurable function \(f\) on \(\mathbb{R}^{n}\), the parametric Marcinkiewicz integral \(\mu_{\Omega}^{\rho}\) related to the Littlewood-Paley \(g\)-function is defined by setting, for all \(x\in\mathbb{R}^{n}\), \[ \mu^{\rho}_{\Omega}(f)(x):=(\int_{0}^{\infty}|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}f(y){dy}|^{2}\frac{dt}{t^{2\rho+1}})^{1/2}, \] where \(\Omega\) is homogeneous of degree zero satisfying the cancellation condition.
In this article, we discuss the boundedness of the parametric Marcinkiewicz integral \(\mu_{\Omega}^{\rho}\) with rough kernel from weak Musielak-Orlicz Hardy space \({WH}^{\varphi}(\mathbb{R}^{n})\) to weak Musielak-Orlicz space \({WL}^{\varphi}(\mathbb{R}^{n})\). These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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