Relating different conditions for the positivity of the Schrödinger operator
type: Item
instance of: scholarly article
title: Relating different conditions for the positivity of the Schrödinger operator
zbMATH Open document ID: 0779.35029
author: Caroline Sweezy
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-01-23T00:00:00Z
review text: Let \(L=-\Delta-(1/c)v\) be a Schrödinger operator in \(\mathbb{R}^ d\). The author shows that the Fefferman condition \[ \left({1\over {| Q|}} \int_ Q v^ p\right)^{1/p} \leq {c \over {\ell^ 2(Q)}},\qquad p>1,\;Q:\text{ cube}, \] where \(\ell(Q)\) is the side length of \(Q\), implies the Kerman-Sawyer condition \[ \int_ Q M^ 2_ 1(\chi_ Q v)\leq c\int_ Q v, \qquad \text{for all }Q, \] where \[ M_ 1(f)(x)=\sup_{{{x\in Q} \atop {Q\text{ any cube }}}} {1 \over {| Q|^{1-1/d}}} \int_ Q f(y)dy, \] which is a necessary and sufficient condition for the positivity of the operator \(L\). He also shows that the Chang, Wilson and Wolff condition \[ {1\over {| Q|}} \int_ Q \ell^ 2(Q)v(y)\log^{2+\varepsilon} (1+\ell^ 2(Q)v(y))dy \leq c \] implies the Kerman-Sawyer condition with \(\int_ Q M^ 2_ 1 (\chi_ Q v)\) replaced by \(\int_{\mathbb{R}^ d} M^ 2_ 1(\chi_ Q v)\). The author's proof is direct and neither \(A_ \infty\) weights nor the Haar-type decomposition is used.
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 218650
zbMATH Keywords: dyadic decomposition
zbMATH Keywords: Orlicz norm
zbMATH Keywords: Hardy-Littlewood maximal function
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Fefferman condition
zbMATH Keywords: Kerman-Sawyer condition
zbMATH Keywords: positivity
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072628
OpenAlex ID: W2078498891
cites work: Weights and L log L
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DOI: 10.1216/RMJM/1181072628
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description: scientific article; zbMATH DE number 218650