Singular integral operators along surfaces of revolution
type: Item
instance of: scholarly article
title: Singular integral operators along surfaces of revolution
zbMATH Open document ID: 1022.42008
DOI: 10.1016/S0022-247X(02)00316-5
author: Hung Viet Le
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: In this paper, the author obtains the boundedness of certain singular integral operators along surfaces with highly singular kernels. The singular integral operator studied in this paper is defined, for \(\alpha,\beta> 0\), by \[ T_{\alpha,\beta}f(x,x_n)= \text{p.v. }\int_{\mathbb{R}^n}f(x- y,x_n- \gamma(|y|)) e^{i|y|^{-\beta}}h(|y|) \Omega(y)|y|^{-n-\alpha} dy, \] where \(x,y\in \mathbb{R}^n\), \(x_n\in \mathbb{R}\), \(\Omega\in L^1(S^{n-1})\) is homogeneous of degree zero and satisfies mean zero over the unit sphere \(S^{n-1}\), \(h\) is a bounded, differentiable radial function, \(\gamma\) is a measurable even function such that \(|\gamma'(r)|\) is increasing on \((\text{supp} \gamma')\cap [0,\infty)\). Assume that either \(h\) is monotone or \(h'\in L^1(\mathbb{R})\) and that \(\gamma'\in L^1(\mathbb{R})\) or \(\gamma\in L^\infty(\mathbb{R})\) and \(\gamma(r)\) is monotone on \([0,\infty)\). The author proves that if \(\Omega\) is a function in the Hardy space \(H^1(S^{n-1})\) then \(T_{\alpha,\beta}\) is bounded on \(L^2(\mathbb{R}^{n+1})\) provided \(0< 2\alpha< \beta\). He also proves that \(T_{\alpha,\beta}\) is bounded on \(L^p(\mathbb{R}^{n+1})\) provided \(\beta/(\beta-\alpha)< p< \beta/\alpha\) with \(0< 2\alpha< \beta\) and that the maximal function \[ M^\gamma g(s)= \sup_{r> 0} r^{-1} \int_{|t|< r}|g(s- \gamma(t))|dt \] is bounded on \(L^p(\mathbb{R}^n)\) for \(1< p< \infty\). Several applications are also given.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 1866648
zbMATH Keywords: surface of revolution
zbMATH Keywords: \(L^p\)-boundedness
zbMATH Keywords: singular integral operators along surfaces
zbMATH Keywords: highly singular kernels
zbMATH Keywords: maximal function
reviewed by: Dashan Fan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866648