On rough Marcinkiewicz integrals along surfaces
type: Item
instance of: scholarly article
title: On rough Marcinkiewicz integrals along surfaces
zbMATH Open document ID: 1202.42033
author: Jiankai Xu
author: Huoxiong Wu
publication date: 2010-04-23T00:00:00Z
review text: Let \(\mathbb R^n\), \(n\geq2\), be the \(n\)-dimensional Euclidean space and \(S^{n-1}\) the unit sphere in \(\mathbb R^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Let \(\Omega\) be a homogeneous function of degree zero on \(\mathbb R^n\) with \(\Omega\in L^1(S^{n-1})\) and \[ \int_{S^{n-1}}\Omega(y')\,d\sigma(y')=0,\tag \(*\) \] where \(x'\equiv x/|x|\) for any \(x\neq0\). For a measurable real-valued function \(b\) on \(\mathbb R^+\), one says that \(b\in\Delta_\nu\), \(\nu>1\), if \[ \|b\|_{\Delta_\nu}\equiv\sup_{R>0}\left(R^{-1}\int_0^R |b(t)|^\nu\,dt\right)^{1/\nu}<\infty. \] For \(m>1\), let \(\Gamma(t)\equiv(\gamma_1(t),\cdots,\gamma_m(t))\) be a curve on \(\mathbb R^m\), where each \(\gamma_j\) is a real-valued continuous function on \(\mathbb R^+\). For \(x\in\mathbb R^n\) and \(x_* \in\mathbb R^m\), define the Marcinkiewicz integral operator \(\mu_{\Omega,b,\Gamma}\) in \(\mathbb R^{n+m}\) along \(\Gamma\) by \[ \mu_{\Omega,b,\Gamma}(f)(x,x_*)\equiv\left(\int_0^\infty\left| F_{t,\Gamma}^b(x,x_*)\right|^2t^{-3}\,dt\right)^{1/2}, \] where \[ F_{t,\Gamma}^b(x,x_*)\equiv\int_{\{y\in\mathbb R^m:\,|y|<t\}}b(|y|) \Omega(y')|y|^{1-n}f(x-y,x_*-\Gamma(|y|))\,dy. \] Also, define the lower dimensional maximal function \(M_\Gamma\) by setting, for all \(x_1\in{\mathbb R}\) and \(x_* \in\mathbb R^m\), \[ M_\Gamma(h)(x_1,x_*)\equiv\sup_{R>0}R^{-1}\int_0^R|h(x_1-t,x_*-\Gamma(t))|\,dt. \] Let \(H^1(S^{n-1})\) denote the Hardy space on the unit sphere. In this paper, the authors prove that if \(n\geq2\), \(m\geq1\), \(b\in\Delta_\nu\) for \(\nu>1\), and \(\Omega\in H^1(S^{n-1})\) satisfies (\(\ast\)), then there exists a positive constant \(C\), independent of \(f\), such that {\parindent=7mm \begin{itemize}\item[(i)] \(\|\mu_{\Omega,b,\Gamma}(f)\|_{L^2(\mathbb R^{n+m})}\leq C \|f\|_{L^2(\mathbb R^{n+m})}\), \item[(ii)] \(\|\mu_{\Omega,b,\Gamma}(f)\|_{L^p(\mathbb R^{n+m})}\leq C \|f\|_{L^p(\mathbb R^{n+m})}\) \end{itemize}} for \(|1/p-1/2|<\min\{1/2,\;1/…
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B99
zbMATH DE Number: 5700799
zbMATH Keywords: Marcinkiewicz integral
zbMATH Keywords: surface
zbMATH Keywords: maximal operator
zbMATH Keywords: Hardy space
zbMATH Keywords: rough kernel
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: s10114-010-7457-4
OpenAlex ID: W2401768369
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DOI: 10.1007/S10114-010-7457-4
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5700799