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.


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.)
Full Text: DOI Euclid


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