Marcinkiewicz integral operators along twisted surfaces
type: Item
instance of: scholarly article
title: Marcinkiewicz integral operators along twisted surfaces
zbMATH Open document ID: 1494.42012
published in: Communications on Pure and Applied Analysis
publication date: 2022-01-21T00:00:00Z
review text: In this paper, the author defined a class of Marcinkiewicz integral operators \(\mathcal{M}_{\Gamma_{P,Q},\Omega,h}\) on product domains by translates determined by twisted surfaces. More precisely, \(\mathcal{M}_{\Gamma_{P,Q},\Omega,h}\) is defined by \[\mathcal{M}_{\Gamma_{P,Q},\Omega,h}f(x,y)=\bigg(\iint_{\mathbb{R}\times\mathbb{R}}|F_{t,s,\Gamma_\alpha,\Omega,h}f(x,y)|^2\frac{dtds}{2^{2t}2^{2s}}\bigg)^{1/2},\] where \[F_{t,s,\Gamma_{P,Q},\Omega,h}f(x,y)=\int_{|u|\le2^t}\int_{|v|\le2^s}f((x,y)-\Gamma_{P,Q}(u,v))\frac {\Omega(u,v)h(|u|,|v|)}{u^{n-1}|v|^{m-1}}dudv.\] Further, \(h\in L^\infty(\mathbb{R}_+\times\mathbb{R}_+)\), \(\Gamma_{P,Q}=(P(|y|)x,Q(|x|)y)\) with \(P,Q:\mathbb{R}\rightarrow\mathbb{R}\) are polynomials and \(\Omega\in L^1(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})\) satisfying: \[\Omega(tx,sy)=\Omega(x,y)\ \text {for any }\ t,s>0,\] \[\int_{\mathbb{S}^{n-1}}\Omega(u^\prime,\cdot)d\sigma_n(u^\prime)=\int_{\mathbb{S}^{m-1}}\Omega(\cdot,v^\prime)d\sigma_m(v^\prime)=0.\] With the above definitions, in this paper, the author gives the \(L^p(\mathbb{R}^n\times\mathbb{R}^m)\) boundedness of the Marcinkiewicz integral operator \(\mathcal{M}_{\Gamma_{P,Q},\Omega,h}\) when \(\Omega\in L(\log^+L)(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})\) and the polynomials \(P,Q\) satisfy some conditions. To prove the above result, the author introduces a two-parameter maximal operator and gives also the \(L^p(\mathbb{R}^n\times\mathbb{R}^m)\) boundedness of this maximal operator.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 7461704
zbMATH Keywords: Marcinkiewicz integral operators
zbMATH Keywords: Fourier transform
zbMATH Keywords: rough kernels
zbMATH Keywords: product domains
author: Ahmad Al-Salman
reviewed by: Yong Ding
MaRDI profile type: Publication
full work available at URL: cpaa.2021173
OpenAlex ID: W3203633051
cites work: Marcinkiewicz integrals on product spaces
cites work: Rough Marcinkiewics integral operators
cites work: Singular integrals on product domains
cites work: Singular integral operators on product domains along twisted surfaces
cites work: Maximal functions along surfaces on product domains
cites work: Rough Marcinkiewicz integrals on product spaces
cites work: Maximal operators with rough kernels on product domains
cites work: Rough singular integrals on product spaces
cites work: Singular Integrals with Rough Kernels in L log L (S n −1 )
cites work: \(L^p\) bounds for the function of Marcinkiewicz.
cites work: Integral operators of Marcinkiewicz type
cites work: CONVOLUTION OPERATORS ON BANACH SPACE VALUED FUNCTIONS
cites work: Rough Marcinkiewicz integrals with \(L(\log^+L)^2\) kernels on product spaces
cites work: Marcinkiewicz integrals with rough homogeneous kernels of degree zero in 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: \(L^2\)-boundedness of a Marcinkiewicz integral with rough kernel
cites work: Singular integral operators with rough kernels supported by subvarieties
cites work: Singular integrals on product spaces
cites work: Harmonic analysis on nilpotent groups and singular integrals. I: Oscillatory integrals
cites work: Q5749786
cites work: On the Functions of Littlewood-Paley, Lusin, and Marcinkiewicz
cites work: BOUNDEDNESS OF MULTIPLE MARCINKIEWICZ INTEGRAL OPERATORS WITH ROUGH KERNELS
DOI: 10.3934/CPAA.2021173
description: scientific article; zbMATH DE number 7461704