Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type
type: Item
instance of: scholarly article
title: Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type
zbMATH Open document ID: 0991.43004
DOI: 10.1007/BF02513077
author: Yongsheng Han
author: Shanzhen Lu
author: Dachun Yang
published in: The Journal of Fourier Analysis and Applications
publication date: 2002-08-24T00:00:00Z
full work available at URL: https://eudml.org/doc/59699
review text: Let \((X, d, \mu)\) be a space of homogeneous type, \(\roman{i.e.}\), \(X\) is a topological space, defined by a pseudometric \(d\), and \(\mu\) is a positive Borel measure on \(X\), such that \(d(x,y)=d(y,x)\geq 0\) and \(=0\) iff \(x=y\), \(d(x,z)\leq K(d(x,y)+d(y,z))\), and \(\mu(B(x,2t))\leq A \mu(B(x,t))\). Here, \(B(x,t)=\{y\in X; d(x,y)<t\}\). The authors treat the space \((X, d, \mu)\) of homogeneous type, satisfying \(|d(x,y)-d(x',y)|\leq Cr^{1-\theta}d(x,x')^\theta\) if \(d(x,y), d(x',y)<r\), and \(A_1r\leq \mu(B_d(x,r))\leq A_2r\) for every \(x\in X\) and \(r>0\). A nonnegative function \(\varphi(t)\) defined on \((0, \infty)\) is said to be of lower (upper) type \(\alpha \geq 0\) if \(\varphi(st)\leq(\geq) C_1s^\alpha \varphi(t)\) for \(0<s\leq 1\) and \(t>0\). \quad In the context of spaces of homogeneous type, G. David, J.L. Journé and S. Semmes showed how to construct an appropriate family of operators \(\{D_k\}_{k\in \mathbb Z}\) whose kernels satisfy certain size, smoothness and moment conditions and \(\sum_{k\in \mathbb Z}D_k=I\) on \(L^2\). Y. Han and E. Sawyer introduced a class of distributions on spaces of homogeneous type and established a Calderón-type reproducing formula for this class, associated with the family of the operators mentioned above. The authors use inhomogeneous approximations to the identity \(\{S_k\}_{k\in \mathbb N}\), defined by \textit{Y. Han} [J. Geometric Anal. 7, 259-284 (1997; Zbl 0915.42010)], and give a Littlewood-Paley theorem for a Littlewood-Paley \(g\)-function associated with this approximation to the identity. Based on it, they give inhomogeneous discrete Calderón-type reproducing formulas.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 1757475
zbMATH Keywords: spaces of homogeneous type
zbMATH Keywords: Calderón reproducing formula
zbMATH Keywords: inhomogeneous
zbMATH Keywords: Littlewood-Paley theorem
zbMATH Keywords: \(g\)-functions
zbMATH Keywords: Calderón-Zygmund operators
zbMATH Keywords: frames
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1757475