Classes of weights related to Schrödinger operators
type: Item
instance of: scholarly article
title: Classes of weights related to Schrödinger operators
zbMATH Open document ID: 1203.42029
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-10-25T00:00:00Z
review text: The authors consider the Schrödinger operator on \(\mathbb R^d\) for \(d\geq 3\), \(\mathcal L=-\Delta + V\), where \(V\) is a non-negative potential satisfying the reverse Hölder inequality \((\frac{1}{|B|}\int_B V^q)^{1/q}\leq C \frac{1}{|B|}\int_B V\) for some \(q>d/2\). The paper concerns the study of weighted \(L^p\)-estimates for several operators associated to \(\mathcal L\), namely the maximal operator of the diffusion semigroup \(\mathcal T^*f(x)=\sup_{t>0} e^{-t\mathcal L}f(x)\), the \(\mathcal L\)-fractional operator \(\mathcal L^{-\alpha/2}\), the \(\mathcal L\)-Riesz transform \(\mathcal R=\nabla \mathcal L^{-1/2}\) or the \(\mathcal L\)-square function. While in the case \(V=0\) the weighted theory depends on the class of \(A_p\)-weights, they show that there is a new class appearing in this context which fits with the boundedness of those operators. They introduce the class \(A^{\rho, \theta}_p\) of those weights \(w\) such that \[ (\int_B w)^{1/p}(\int_B w^{-1/(p-1)})^{1/p'}\leq C |B|(1+\frac{r}{\rho(x)})^\theta \] for every ball \(B=B(x,r)\), where \(\rho(x)\) is the critical radius introduced by \textit{Z. Shen} [Ann. Inst. Fourier 45, No. 2, 513--546 (1995; Zbl 0818.35021)] given by \(\rho(x)= \sup\{ r: \frac{1}{r^{d-2}}\int_{B(x,r)} V\leq 1\} \). They show that if \(w\in A_p^{\rho,\infty}=\cup_{\theta \geq 0}A^{\rho, \theta}_p\) for \(1<p<\infty\) then the maximal operator \(\mathcal T^*\) and the \(\mathcal L\)-square function operator are bounded in \(L^p(w)\). Also the weak-type result holds in the case \(w\in A_1^{\rho,\infty}\). The boundedness for the \(\mathcal L\)-Riesz transform is shown under the same assumptions on the weight \(w\), but only for \(V\) satisfying the revere Hölder inequality for \(q\geq d\). Finally, the expected result on the \(\mathcal L\)-fractional operator is also achieved.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 5804530
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Riesz transform
zbMATH Keywords: maximal operator
zbMATH Keywords: weight
author: B. Bongioanni
author: E. Harboure
author: O. Salinas
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.08.008
OpenAlex ID: W2035181034
cites work: Riesz transforms related to Schrödinger operators acting on BMO type spaces
cites work: Weighted norm inequalities for maximal functions and singular integrals
cites work: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality
cites work: Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality
cites work: Q3700300
cites work: An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Uniformly Elliptic Operators with Non-Negative Potentials
cites work: \(L^ p\) estimates for Schrödinger operators with certain potentials
DOI: 10.1016/J.JMAA.2010.08.008
description: scientific article; zbMATH DE number 5804530