Maximal functions along surfaces in product spaces
type: Item
instance of: scholarly article
title: Maximal functions along surfaces in product spaces
zbMATH Open document ID: 1090.42011
author: Hung Viet Le
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: Let \(\Gamma: \mathbb R^n \times \mathbb R^m \mapsto \mathbb R\) be a measurable function which is radial with respect to both variables, that is, \[ \Gamma(y_1, y_2)= \Gamma(|y_1|, |y_2|)\qquad \text{for all}\;y_1\in \mathbb R^n\;\text{and}\;y_2\in \mathbb R^m. \] The author investigates the \(L^p\)-boundedness of the maximal operator \[ \begin{multlined} M_\Gamma f(x_1, x_2, x_3)= \\ \sup_{r_1, r_2 >0}\bigg\{\frac 1{r_1^n r_2^m} \int_{|y_2|\leq r_2} \int_{|y_1|\leq r_1} \big|f\big(x_1-y_1, x_2-y_2, x_3-\Gamma(|y_1|, |y_2|)\big)\big|\, dy_1 dy_2\bigg\},\end{multlined} \] where \(x_1, y_1 \in \mathbb R^n\), \(x_2, y_2 \in \mathbb R^m\), and \(x_3\in \mathbb R\). Applying the result in [\textit{J. Duoandikoetxea}, Ann. Inst. Fourier 36, No. 4, 185--206 (1986; Zbl 0568.42011)], the author proves that, under certain conditions on \(\Gamma\), the operator \(M_\Gamma\) is bounded on \(L^p(\mathbb R^n \times \mathbb R^m \times \mathbb R)\) for \(p>1\) and \(n, m \geq 1\).
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5016191
zbMATH Keywords: maximal operator
zbMATH Keywords: Hardy--Littlewood maximal function
zbMATH Keywords: product space
zbMATH Keywords: \(L^p\)-boundedness
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.04.054
OpenAlex ID: W2030340491
cites work: Estimates for maximal functions along hypersurfaces
cites work: Multiparameter Maximal Functions Along Dilation-Invariant Hypersurfaces
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 Hilbert Transforms Along Surfaces and Convolution Operators Homogeneous with Respect to a Multiple Parameter Group
cites work: Problems in harmonic analysis related to curvature
cites work: Singular integrals supported on submanifolds
cites work: \(L^ p\)-boundedness of the multiple Hilbert transform along a surface
DOI: 10.1016/J.JMAA.2005.04.054
Recommended article: Maximal functions along surfaces on product domains
Recommended article: MAXIMAL FUNCTIONS ALONG TWISTED SURFACES ON PRODUCT DOMAINS
Recommended article: Maximal functions associated to surfaces of revolution on product domains
Recommended article: On maximal operators along surfaces
Recommended article: Q4972523
Recommended article: Bilinear maximal functions associated with surfaces
Recommended article: \(L^p\) estimates for maximal functions along surfaces of revolution on product spaces
Recommended article: Maximal functions over hypersurfaces with flat points.
Recommended article: Maximal functions, product condition and its eccentricity
Recommended article: Estimates for maximal functions along hypersurfaces
description: scientific article; zbMATH DE number 5016191