On local oscillatory integrals with variable Calderón-Zygmund kernels
type: Item
instance of: scholarly article
title: On local oscillatory integrals with variable Calderón-Zygmund kernels
zbMATH Open document ID: 0933.42007
DOI: 10.1007/BF01291837
author: Zusheng Zhou
author: Shanzhen Lu
author: Dachun Yang
published in: Integral Equations and Operator Theory
publication date: 2000-03-21T00:00:00Z
review text: Let \(p\in (1,\infty)\). In the paper, the authors study the uniform \(L^p(\mathbb{R}^n)\) boundedness in \(\lambda\) of the oscillatory singular integral operator \(T_\lambda\) defined by \[ T_\lambda f(x)= \text{p.v. }\int_{\mathbb{R}^n} e^{i\lambda\Phi(x, y)} K(x,x- y) \psi(x, y) f(y) dy, \] where \(\lambda\in \mathbb{R}\), \(\Phi\) is a real analytic function or a real \(C^\infty\) function on \(\mathbb{R}^n\times \mathbb{R}^n\), \(\psi\in C^\infty_0(\mathbb{R}^n\times \mathbb{R}^n)\) and \(K\) is a variable Calderón-Zygmund kernel. Moreover, after obtaining the weighted \(L^p\) boundedness, the authors also study the uniform boundedness in \(\lambda\) of the commutators generated by \(T_\lambda\) and \(\text{BMO}(\mathbb{R}^n)\) functions. In the proof, the authors assume that the kernel \(K\) satisfies the condition \[ \max_{|j|\leq 2n} \Biggl\|{\partial^{|j|}\over\partial y^j} K(x,y)\Biggl\|_{L^\infty(\mathbb{R}^n\times S^{n- 1})}= M< \infty. \] So the proof can follow a standard argument by Calderón and Zygmund and an integration by parts yields the necessary estimate. Clearly, the paper would be more interesting if the above condition on the kernel \(K\) can be weakened.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 1310385
zbMATH Keywords: oscillatory singular integral operator
zbMATH Keywords: variable Calderón-Zygmund kernel
zbMATH Keywords: \(L^p\) boundedness
zbMATH Keywords: commutators
reviewed by: Dashan Fan
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01291837
OpenAlex ID: W2052052855
description: scientific article; zbMATH DE number 1310385