NWO sequences, weighted potential operators, and Schrödinger eigenvalues
type: Item
instance of: scholarly article
title: NWO sequences, weighted potential operators, and Schrödinger eigenvalues
zbMATH Open document ID: 0845.42007
DOI: 10.1215/S0012-7094-93-07208-0
author: Richard Rochberg
published in: Duke Mathematical Journal
publication date: 1996-09-04T00:00:00Z
review text: The author applies the theory of nearly weakly orthonormal (NWO) sequences of functions to give sufficient conditions for boundedness of various potential operators between spaces with weights. He applies these results to give eigenvalue estimates for the Schrödinger operator. Let \({\mathbf Q}\) denote the set of all dyadic cubes in \(\mathbb{R}^n\). For each \(Q\in{\mathbf Q}\) let \(\zeta_Q\) denote the center of \(Q\), \(\eta_Q\) denote the sidelength of \(Q\), \(|Q|\) the volume of \(Q\), and denote by \(KQ\) the cube with the same center as \(Q\) but \(K\) times the side length. A set of functions \(\{e_Q(x)\}_{Q\in{\mathbf Q}}\) is an NWO sequence if the associated maximal function \[ f\to f^*(x)= \sup_{|\zeta_Q- x|\leq \eta_Q} (|Q|^{- 1/q}\langle f, e_Q\rangle) \] is bounded on \(L^q\), where \(1/q+ 1/p= 1\). Sufficient conditions for this are that the \(\{e_Q\}\) have support in \(KQ\) and that there exists a \(p< r\leq \infty\) and \(c> 0\) so that for all \(Q\), \(|e_Q|_r\leq c|Q|^{1/r- 1/p}\). Another preliminary result worth mentioning is that it is easy to give estimates and finite rank approximations for operators written as tensor products of NWO sequences of functions. Given a sequence \(\{a_Q\}_{Q\in{\mathbf Q}}\), define a new sequence \(\{a^*_Q\}_{Q\in{\mathbf Q}}\) by \(a^*_Q= \sum_{R\in{\mathbf Q}, R\subset Q} {|R|\over |Q|} |a_R|\), where one should note the typo in the article which writes \(a_Q\) instead of \(a_R\). If the sequence \(\{a^*_Q\}\) is bounded, the discrete Carleson measure norm of the sequence is \(|\{a_Q\}|_{CM_d}= \sup\{a^*_Q\}\). If an operator acting on \(L^p\) can be written as \(T= \sum_{Q\in{\mathbf Q}} a_Q e_Q\otimes f_Q\), where \(\{e_Q\}_{Q\in{\mathbf Q}}\) is NWO for \(L^q\) and \(\{f_Q\}_{Q\in{\mathbf Q}}\) is NWO for \(L^p\), then \(T\) is bounded from \(L^p\) into \(L^p\) and \(|T|\leq C|\{a_Q\}|_{CM_d}\). If \(\{a^*_Q\}\) tends to zero, one can define the nonincreasing rearrangement of \(\{a^*_Q\}\), which is den…
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 47A80
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 558999
zbMATH Keywords: weighted potential operators
zbMATH Keywords: Schrödinger eigenvalues
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: maximal function
zbMATH Keywords: Riesz potential
zbMATH Keywords: tensor product decomposition
zbMATH Keywords: finite rank approximation
reviewed by: Raymond Johnson
MaRDI profile type: Publication
cites work: Weighted Nonlinear Potential Theory
cites work: Q4011162
cites work: Norm inequalities for potential-type operators
cites work: Weighted Poincare and Sobolev Inequalities and Estimates for Weighted Peano Maximal Functions
cites work: LPEstimates for fractional integrals and sobolev inequalities with applications to schrödinger operators
cites work: Some weighted norm inequalities concerning the Schrödinger operators
cites work: The uncertainty principle
cites work: Q3700300
cites work: The trace inequality and eigenvalue estimates for Schrödinger operators
cites work: Weighted Norm Inequalities for Fractional Integrals
cites work: On a Factor Automorphism of a Normal Dynamical System
cites work: Q4712841
cites work: Q4191141
cites work: Q3349462
cites work: A decomposition theorem for BMO and applications
cites work: Nearly weakly orthonormal sequences, singular value estimates, and Calderon-Zygmund operators
cites work: Q4730335
cites work: Q4730334
cites work: Weighted Inequalities for Fractional Integrals on Euclidean and Homogeneous Spaces
cites work: Weighted Norm Estimates for Sobolev Spaces
cites work: The Spectrum of the Schrodinger Operator
cites work: Q3746171
cites work: Q3857325
cites work: Q5607484
cites work: Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through \(L^{p{\infty}}\)
full work available at URL: s0012-7094-93-07208-0
OpenAlex ID: W1535036118
description: scientific article; zbMATH DE number 558999