On the Fefferman-Phong inequality

Boulkhemair, Abdesslam. "On the Fefferman-Phong inequality." Annales de l’institut Fourier 58.4 (2008): 1093-1115. <http://eudml.org/doc/10343>.

@article{Boulkhemair2008,
abstract = {We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by $\{n\over 2\}+4+\epsilon $ improving thus the bound $2n+4+\epsilon $ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type $S^0_\{0,0\}$, we show that this number is bounded by $n+4+\epsilon $; more precisely, for a non negative symbol $a$, the Fefferman-Phong inequality holds if $\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )$ are bounded for, roughly, $4\le |\alpha |+|\beta |\le n+4+\epsilon $. To obtain such results and others, we first prove an abstract result which says that the Fefferman-Phong inequality for a non negative symbol $a$ holds whenever all fourth partial derivatives of $a$ are in an algebra $\{\mathcal\{A\}\}$ of bounded functions on the phase space, which satisfies essentially two assumptions : $\{\mathcal\{A\}\}$ is, roughly, translation invariant and the operators associated to symbols in $\{\mathcal\{A\}\}$ are bounded in $L^2$.},
affiliation = {Université de Nantes Laboratoire de Mathématiques Jean Leray CNRS UMR6629 2, rue de la Houssinière BP 92208 44322 Nantes (France)},
author = {Boulkhemair, Abdesslam},
journal = {Annales de l’institut Fourier},
keywords = {Fefferman-Phong inequality; Gårding inequality; symbol; $S^m_\{\varrho ,\delta \}$; pseudodifferential operator; Weyl quantization; Wick quantization; semi-boundedness; $L^2$ boundedness; algebra of symbols; uniformly local Sobolev space; Hölder space; semi-classical; Weyl-Hörmander class; boundedness},
language = {eng},
number = {4},
pages = {1093-1115},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Fefferman-Phong inequality},
url = {http://eudml.org/doc/10343},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Boulkhemair, Abdesslam
TI - On the Fefferman-Phong inequality
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 4
SP - 1093
EP - 1115
AB - We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by ${n\over 2}+4+\epsilon $ improving thus the bound $2n+4+\epsilon $ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type $S^0_{0,0}$, we show that this number is bounded by $n+4+\epsilon $; more precisely, for a non negative symbol $a$, the Fefferman-Phong inequality holds if $\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )$ are bounded for, roughly, $4\le |\alpha |+|\beta |\le n+4+\epsilon $. To obtain such results and others, we first prove an abstract result which says that the Fefferman-Phong inequality for a non negative symbol $a$ holds whenever all fourth partial derivatives of $a$ are in an algebra ${\mathcal{A}}$ of bounded functions on the phase space, which satisfies essentially two assumptions : ${\mathcal{A}}$ is, roughly, translation invariant and the operators associated to symbols in ${\mathcal{A}}$ are bounded in $L^2$.
LA - eng
KW - Fefferman-Phong inequality; Gårding inequality; symbol; $S^m_{\varrho ,\delta }$; pseudodifferential operator; Weyl quantization; Wick quantization; semi-boundedness; $L^2$ boundedness; algebra of symbols; uniformly local Sobolev space; Hölder space; semi-classical; Weyl-Hörmander class; boundedness
UR - http://eudml.org/doc/10343
ER -