\(Tb\) theorem for the generalized singular integral operator on product Lipschitz spaces with para-accretive functions
type: Item
instance of: scholarly article
title: \(Tb\) theorem for the generalized singular integral operator on product Lipschitz spaces with para-accretive functions
zbMATH Open document ID: 1454.42014
author: Taotao Zheng
author: Xiangxing Tao
published in: The New York Journal of Mathematics
publication date: 2020-10-08T00:00:00Z
full work available at URL: 26-43.html
review text: Spaces \(\mathrm{Lip}_b(\alpha_1,\alpha_2)(\mathbb{R}^n\times\mathbb{R}^m)\) are defined for product \(b=b_1b_2\) para-accretive functions and a \(Tb\) theorem for generalized singular integral operators on these spaces is established. A function \(b\) defined on \(\mathbb{R}^n\) is para-accretive if there are \(C>0\) and \(\delta>0\) such that for any cube \(Q\subset\mathbb{R}^n\) there is a subcube \(Q^\prime\) such that \(\delta|Q|\leq |Q^\prime|\) and \(\frac{1}{|Q|}\Bigl| \int_{Q^\prime} b(x)\, dx\Bigr|\geq C\). The space \(\mathrm{Lip}(\alpha_1,\alpha_2)(\mathbb{R}^n\times\mathbb{R}^m)\) consists of those \(f\) such that \[|f(x-u,y-v)-f(x,y-v)-f(x-u,y)+f(x,y)|\leq C|u|^{\alpha_1}|v|^{\alpha_2} \] (with norm defined as the smallest \(C\) such that the inequality holds for all \((x,y)\in (\mathbb{R}^n\times\mathbb{R}^m\)). One defines \(\mathrm{Lip}_b(\alpha_1,\alpha_2)=\{f: fb\in\mathrm{Lip}(\alpha_1,\alpha_2)\}\) where \(b(x,y)=b_1(x)b_2(y)\). One defines \(C_0^\eta(\mathbb{R}^n)\) to be those compactly supported \(f\) such that \(\sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\eta}<\infty\). One defines a corresponding product space with product norms in terms of the differences defined above. One defines a product Calderón-Zygmund operator in terms of a product kernel \(K(x_1,y_1;x_2,y_2)\) satisfying standard CZO estimates in each parameter along with additional joint decay estimates on cross-parameter differences. One defines a general singular integral operator \(T\) as a continuous linear operator from \(b C_0^\eta(\mathbb{R}^n\times\mathbb{R}^m)\) into \((b C_0^\eta(\mathbb{R}^n\times\mathbb{R}^m))^\prime\) associated with a product CZO kernel such that the formal integrals \(\langle M_b T M_b f_1\otimes f_2, \, g_1\otimes g_2\rangle\) are defined by product integration against the kernel for Hölder continuous test functions \(f_i\), \(g_i\) having disjoint supports. One also defines test functions of type \((\beta_1,\beta_2,\gamma_1,\gamma_2)\) centered at \((…
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 7258098
zbMATH Keywords: product homogeneous Lipschitz spaces
zbMATH Keywords: Littlewood-Paley theory
zbMATH Keywords: generalized singular integral operator
zbMATH Keywords: Besov spaces
zbMATH Keywords: para-accretive function
reviewed by: Joseph Lakey
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7258098