Marcinkiewicz integrals with rough kernels on \(L^p(1\leq p\leq 2)\)
type: Item
instance of: scholarly article
title: Marcinkiewicz integrals with rough kernels on \(L^p(1\leq p\leq 2)\)
zbMATH Open document ID: 0923.42012
author: Kyung Soo Rim
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-10-12T00:00:00Z
review text: Let \(n\geq 3\) and \(H\in L^\infty(S^{n-1})\) be a homogeneous function of positive degree \(\leq n-1\) satisfying \(\int_{S^{n-1}}H(x') dx'=0,\) and let \(\phi\) be a characteristic function on a compact set. Define \(\Omega =H\phi\), \[ F_t(x)=\int_{| x-y| \leq t} {\Omega(x-y)\over {| x-y| ^{n-1}}}f(y) dy \] and \[ \mu_\Omega(f)(x)=\left(\int^\infty_0| F_t(x)| ^2 {dt\over t^3}\right)^{1/2}, \] which is called to be the generalized Marcinkiewicz integral. The author proves that \(\mu_\Omega\) is bounded on \(L^p({\mathbf{R}}^n)\) for \(1<p\leq 2\) and of weak type \((1,1)\).
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1286153
zbMATH Keywords: Marcinkiewicz integral
zbMATH Keywords: rough kernel
zbMATH Keywords: boundedness
zbMATH Keywords: weak type \((1,1)\)
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6276
OpenAlex ID: W2025542436
cites work: CONVOLUTION OPERATORS ON BANACH SPACE VALUED FUNCTIONS
cites work: On a Theorem of Marcinkiewicz and Zygmund
cites work: On the existence of certain singular integrals
cites work: The Marcinkiewicz integral operator with Hardy space function kernel on product domains
cites work: Multiple singular integrals and maximal functions along hypersurfaces
cites work: Maximal and singular integral operators via Fourier transform estimates
cites work: A class of singular integral operators with rough kernel on product domains
cites work: On certain maximal functions and approach regions
cites work: Marcinkiewicz integral operators on product domains
cites work: On the Functions of Littlewood-Paley, Lusin, and Marcinkiewicz
cites work: Q5607484
cites work: Q3142876
cites work: A note on the Marcinkiewicz integral
cites work: On Certain Integrals
DOI: 10.1006/JMAA.1999.6276
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description: scientific article; zbMATH DE number 1286153