Maximal operators and singular integrals with non-isotropic dilation on product domains
type: Item
instance of: scholarly article
title: Maximal operators and singular integrals with non-isotropic dilation on product domains
zbMATH Open document ID: 1214.42022
author: Zhongkai Li
author: Huoxiong Wu
author: Bolin Ma
publication date: 2010-10-29T00:00:00Z
review text: Let \(n_{1},n_{2}\geq 2\) and let \(P_{j}\) be a real \(n_{j}\times n_{j}\) matrix whose eigenvalues have positive real parts. Let \(\gamma _{j}\) \(=\) trace \(P_{j}\) for \(j\in \{1,2\}.\) For \(j\in \{1,2\},\) define a dilation group \(\{A_{j,t}\}_{t>0}\) on \(\mathbb{R}^{n_{j}}\) by \(A_{j,t}=t^{P_{j}}=\exp (P_{j}\log t).\) Then there exists a non-negative function \(\rho _{j}\) on \( \mathbb{R}^{n_{j}}\) associated with \(\{A_{j,t}\}_{t>0}\) such that \(\rho _{j}\) is continuous on \(\mathbb{R}^{n_{j}}\) and infinitely differentiable \ on \( \mathbb{R}^{n_{j}}\backslash \{0\}\) and satisfies the following: (i) \(\rho _{j}(A_{j,t}x_{j})=t\rho _{j}(x_{j})\) for all positive values of \(t\) and \( x_{j}\in \mathbb{R}^{n_{j}};\) (ii) \(\rho _{j}(x_{j}+y_{j})\leq C\left( \rho _{j}(x_{j})+\rho _{j}(y_{j})\right) \) for all \(x_{j},y_{j}\in \mathbb{R} ^{n_{j}}\) and for some positive constant \(C;\) (iii) If \(\sum_{j}=\left\{ x_{j}:\rho _{j}(x_{j})=1\right\} ,\) then \(\sum_{j}=\left\{ \theta _{j}:\left\langle B_{j}\theta _{j},\theta _{j}\right\rangle )=1\right\} \) for a positive symmetric matrix \(B_{j}\). Then the Lebesgue measure \( dx_{j}=t^{^{\gamma _{j}}-1}d\sigma _{j}dt.\) That is, \[ \int_{\mathbb{R}^{n_{j}}}f(x_{j})dx_{j}=\int_{0}^{\infty }\int_{\sum_{j}}f(A_{j,t}\theta _{j})d\sigma _{j}(\theta _{j})dt \] for appropriate functions \(f,\) where \(d\sigma _{j}\) is a \(C^{\infty }\) \ measure on \(\sum_{j},\) \(j\in \{1,2\}\). Let \(\Omega \in L^{1}(\sum_{1}\times \sum_{2})\) and let \(\Omega\) satisfy \(\Omega (A_{1,t_{1}}x_{1},A_{2,t_{2}}x_{2})=\Omega (x_{1},x_{2})\) for all \( t_{1},t_{2}>0\) and \[ \int_{\sum_{1}}\Omega (\theta _{1},\theta _{2})d\sigma _{1}(\theta _{1})=\int_{\sum_{2}}\Omega (\theta _{1},\theta _{2})d\sigma _{2}(\theta _{2})=0. \] Define the non-isotropic singular integral operator \(T_{\Omega ,h}\) on \( \mathbb{R}^{n_{1}}\times \mathbb{R}^{n_{2}}\) by \[ T_{\Omega ,h}f(x_{1},x_{2})=\text{p.v.}\int \int_{\mathbb{R}^{n_{1}}\times \mathbb{…
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 5808617
zbMATH Keywords: non-isotropic singular integral
zbMATH Keywords: maximal function
zbMATH Keywords: product domain
zbMATH Keywords: rough kernel
reviewed by: Hussain Al-Qassem
MaRDI profile type: Publication
full work available at URL: s10114-010-8400-4
OpenAlex ID: W2067190913
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DOI: 10.1007/S10114-010-8400-4
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5808617