Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators
type: Item
instance of: scholarly article
title: Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators
zbMATH Open document ID: 1172.47018
author: Yuan Zhou
author: Dachun Yang
author: Dongyong Yang
published in: Potential Analysis
publication date: 2009-06-17T00:00:00Z
review text: This is a highly elaborated paper, touching several topics from the theory of singular integrals, with sound applications to harmonic analysis, Riesz transform, Hardy function spaces and related concepts. In order to give a flavor of the material, we shall provide a sample result, considered by the authors, to constitute one of the main results of the paper. First, an admissible function \(\rho(y)\) is defined on \(\mathbb R^n\), and satisfies on this space the inequality \[ \rho(y)\leq\widetilde C[\rho(x)]^{1/(1+k_0)} [\rho(x)+|x-y|]^{k_0/ (1+k_0)}, \] with \(\widetilde C\) and \(k_0\) positive constants. Then a weighted Hardy space \(H^1_\rho(\mathbb R^n)\) is defined, starting with the classical Hardy space \(H^1(\mathbb R^n)\) and the admissible function \(\rho\). Further on, the operators (Riesz transform) \[ \widetilde R_j(f)(x)\equiv\text{p.v.}\,c_n\int_{\mathbb{R}^n}\frac{x_j-y_j}{|x-y|^{n+1}}\eta\left(\frac{|x-y|}{\rho(x)}\right)f(y)\,dy, \] are considered, as well as their adjoint operators \(R^*_j(f)(x)\). These operators are defined on \(\bigcup^\infty_{p=1}L^p(\mathbb R^n)\). The result can be stated as follows: For any admissible function \(\rho\), the following conditions are equivalent {\parindent=7mm \begin{itemize}\item[(i)] \(f \in H^1_\rho(\mathbb{R}^n)\); \item[(ii)] \(f\in L^1(\mathbb{R}^n)\) and \(\widetilde R_j(f)\in L^1(\mathbb{R}^n)\) for all \(j \in\{1,\dots,n\}\); \item[(iii)] \(f\in L^1(\mathbb{R}^n)\) and \(\widetilde R^*_j (f)\in L^1(\mathbb{R}^n)\) for all \(j\in\{1,\dots,n\}\). \end{itemize}} Moreover, for all \(f\in H^1_\rho(\mathbb{R}^n)\), \[ \|f\|_{H^1_\rho(\mathbb{R}^n)}\sim\|f\|_{L^1 (\mathbb{R}^n)}+\sum^n_{j=1}\|\widetilde R_j(f)\|_{L^1(\mathbb{R}^n)}\sim \|f\|_{L^1 (\mathbb{R}^n)}+\sum^n_{j=1}\|\widetilde R^*_j(f)\|_{L^1 (\mathbb{R}^n)}. \] The meaning of \(f\sim g\) is: \(f\sim g\) iff \(C_1f\leq g\leq C_2f\), with \(C_1,C_2\) positive constants. Several of the paper's results are expressing the continuity (boundedness) of v…
reviewed by: Constantin Corduneanu
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5566067
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Riesz transform
zbMATH Keywords: maximal operator
zbMATH Keywords: adjoint operator
zbMATH Keywords: fractional integral
zbMATH Keywords: admissible function
zbMATH Keywords: Hardy space
zbMATH Keywords: \(\text{BMO}_\rho (\mathbb R^n)\)
zbMATH Keywords: Fefferman-Stein decomposition
zbMATH Keywords: \(\text{BLO}_\rho (\mathbb R^n)\)
MaRDI profile type: Publication
full work available at URL: s11118-009-9116-x
OpenAlex ID: W1975502162
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DOI: 10.1007/S11118-009-9116-X
description: scientific article; zbMATH DE number 5566067