A note on Marcinkiewicz integral operators.
type: Item
instance of: scholarly article
title: A note on Marcinkiewicz integral operators.
zbMATH Open document ID: 1037.42014
DOI: 10.1016/S0022-247X(03)00244-0
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(\mathbb{R}^n\), \(n\geq 2\), be the \(n\)-dimensional Euclidean space and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Let \(\Omega\) be a homogeneous function of degree 0 satisfying \(\Omega\in L^1(S^{n-1})\) and \(\int_{S^{n-1}} \Omega(y')\, d\sigma(y')= 0\), where \(y'= y/| y|\in S^{n-1}\) for any \(y\neq 0\). In this paper the authors study the Marcinkiewicz integral operator defined by \[ {\mathcal M}_\Omega f(z)= \Biggl( \int_0^\infty| K_t* f(x)|^2 t^{-1}\, dt\Biggr)^{1/2}, \] where \(K(x)=| x|^{1-n} \Omega(x) \chi_{(0,1)} (| x|)\), \(K_t(x)= t^{-n} K(x/t)\), and \(\chi_A\) denotes the characteristic function of a set \(A\). Let \(B_q^{\mu,\nu} (S^{n-1})\) be the block space defined in the book by \textit{S. Z. Lu, M. H. Taibleson} and \textit{G. Weiss} [``Spaces generated by blocks'', Beijing Normal University Press, Beijing (1989), per bibl.]. The main Theorem of this paper is the following theorem: (a) If \(\Omega\) belongs to the block space \(B_q^{0,-1/2} (S^{n-1})\), \(q>1\), then \(\|{\mathcal M}_\Omega f\|_p\leq C\| f\|_p\) for any \(p\in (1,\infty)\). (b) There exists an \(\Omega\) which lies in \(B_q^{0,\nu} (S^{n-1})\) for all \(-1< \nu< -1/2\) such that \({\mathfrak M}_\Omega\) is not bounded on \(L^2(\mathbb{R}^n)\).
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1957243
zbMATH Keywords: Marcinkiewicz integrals
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: Fourier transform
zbMATH Keywords: \(L^p\)-boundedness
zbMATH Keywords: rough kernels
zbMATH Keywords: block spaces
author: Hussain Al-Qassem
author: Ahmad Al-Salman
reviewed by: Dashan Fan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1957243