Quantitative weighted estimates for some singular integrals related to critical functions
type: Item
instance of: scholarly article
title: Quantitative weighted estimates for some singular integrals related to critical functions
zbMATH Open document ID: 1482.42017
author: The Quan Bui
author: Xuan Thinh Duong
author: Bui The Anh
published in: The Journal of Geometric Analysis
publication date: 2021-09-02T00:00:00Z
review text: This paper is concerned with quantitative weighted estimates for certain singular integrals corresponding to the new class of weights on the space of homogeneous type. The assumptions on the kernels of these singular integrals do not have any regularity conditions. Recall that \((X, d, \mu)\) is a metric space with the distance \(d\) and a nonnegative Borel doubling measure \(\mu\). The measure \(\mu\) satisfies the doubling property and the reverse doubling property. Quantitative weighted estimates for singular integrals has attracted a lot of attention. \textit{T. P. Hytönen} [Ann. Math. (2) 175, No. 3, 1473--1506 (2012; Zbl 1250.42036)] proved that, if \(T\) is a Calderón-Zygmund operator, Muckenhoupt weights \(w \in A_p\), then \[ \|T\|_{L_{w}^{p} \rightarrow L_{w}^{p}} \leq C_{T, n, p}[w]_{A_{p}}^{\max \left\{1, \frac{1}{p-1}\right\}},\ 1<p<\infty \] and the weighted estimate is sharp. By making use of a supremum of sparse operators, \textit{A. K. Lerner} [Int. Math. Res. Not. 2013, No. 14, 3159--3170 (2013; Zbl 1318.42018)] gave a simpler proof of the above estimate. In order to extend it to some weight class which is larger than the class of Muckenhoupt weights, \textit{B. Bongioanni} et al. [J. Math. Anal. Appl. 373, No. 2, 563--579 (2011; Zbl 1203.42029)] introduced a new class of weights related to the Schrödinger operator \(L\) which contains all the Muckenhoupt weights. This new class of weights is suitable to the study of the boundedness of singular integrals such as the maximal operator, the Riesz transform and the square functions associated to the Schrödinger operator \(L\). The regularity of the heat kernel of \(L\) plays an important role. \textit{F. K. Ly} [J. Math. Soc. Japan 68, No. 2, 489--533 (2016; Zbl 1348.35060)] gave the boundedness of the spectral multiplier and the generalized Riesz transform on weighted \(L^p_w\) spaces with \( w \in {A_{p}^{\rho, \theta}}\). This paper proves the quantitative weighted estimates with respect to new weig…
reviewed by: Taotao Zheng
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 7389669
zbMATH Keywords: critical function
zbMATH Keywords: quantitative weighted estimate
zbMATH Keywords: sparse operator
Wikidata QID: Q114221038
MaRDI profile type: Publication
full work available at URL: s12220-021-00641-0
OpenAlex ID: W3139507494
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DOI: 10.1007/S12220-021-00641-0
description: scientific article; zbMATH DE number 7389669