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Abdesslam Boulkhemair
On the Fefferman-Phong inequality
(Sur l’inégalité de Fefferman-Phong)
Annales de l'institut Fourier, 58 no. 4 (2008), p. 1093-1115, doi: 10.5802/aif.2379
Article PDF | Reviews MR 2427955 | Zbl 1145.35099
Class. Math.: 35Axx, 35Sxx, 47G30, 58J40
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

Résumé - 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 $L^2$.

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