Endpoint estimates for commutators of singular integrals related to Schrödinger operators
type: Item
instance of: scholarly article
title: Endpoint estimates for commutators of singular integrals related to Schrödinger operators
zbMATH Open document ID: 1352.42020
DOI: 10.4171/RMI/871
author: Luong Dang Ky
published in: Revista Matemática Iberoamericana
publication date: 2016-02-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.6335
review text: Let \(V\) be a nonnegative potential that satisfies a reverse Hölder inequality with exponent \(d/2\), where \(d\geq 3\) is the dimension. The class of operators \(\mathcal K_L\) is defined to be the class of sublinear operators \(T\) that map the Hardy space \(H^1 _L(\mathbb R^d)\) associated with \(L\) to \(L^1(\mathbb R^d)\). Here \(H^1 _L\) consists of all functions \(f\in L^1(\mathbb R^d)\) for which \(M_Lf:=\sup_{t>0} |e^{-tL f}|\in L^1(\mathbb R^d)\). For example, by definition, the maximal operator \(M_Lf\) is a \(\mathcal K_L\)-class operator. Of particular importance are the \(L\)-Riesz transforms \(\partial-{x_j} L^{-1/2}\). More generally, the author considers a class of operators which are associated with the Schrödinger operator \(L:=-\Delta+V\) as follows. First one defines a size function \(\rho\) by means of \(\rho(x):=\sup\{r>0:r^{-d+2}\int_{B(x,r)} V(y)dy\leq 1\}\) where \(x\in \mathbb R^d\). An operator \(T\) is a \((\delta,L)\)-Schrödinger-Calderón-Zygmund operator if it is associated with a kernel \(K\) which satisfies the usual \(\delta\)-Hölder continuity condition, in the classical sense, and additionally the size condition \(|K(x,y)|\lesssim_N \frac{1}{|x-y|^d} \big(1+\frac{|x-y|}{\rho(x)}\big)^{-N}\), for every \(N>0\). All \(L\)-Schrödinger-Calderón-Zygmund operators are of \(\mathcal K_L\)-class. The main questions revolve around finding sufficient conditions on symbols \(b\) such that the commutators \([T,b]f:=T(bf)-bT(f)\) map \(H^1 _L\) to \(L^{1,\infty}\) for \(\mathcal K_L\)-class operators \(T\). The main result of the paper shows that if \(T\in\mathcal K_L\) is of weak type \((1,1)\) and \(b\in\mathrm{BMO}(\mathbb R^d)\), then \(T\) maps \(H^1 _L\) into \(L^{1,\infty}\). In particular, this is true for the \(L\)-Riesz transforms. The author concludes the result above by a very precise decomposition of the commutator under the assumptions described before. For example, if \(T\in\mathcal K_L\) is linear and \(b\in\mathrm{BMO}(\mathb…
reviewed by: Ioannis Parissis
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 6539486
zbMATH Keywords: singular integrals
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: Calderón-Zygmund operators
zbMATH Keywords: commutators
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Riesz transforms
zbMATH Keywords: BMO space
zbMATH Keywords: atoms
MaRDI profile type: Publication
arXiv ID: 1203.6335
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description: scientific article