With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Abdesslam BoulkhemairOn 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, 58J40Keywords: 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 - AbstractWe 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 $L^2$. Bibliography[1] J.-M. Bony, Sur l’inégalité de Fefferman-Phong, in Séminaire EDP, Ecole polytechnique, Exposé no 3, 1998–1999 Cedram [2] A. Boulkhemair, “$L^2$ estimates for pseudodifferential operators”, Ann. Sc. Norm. Sup. Pisa IV, XXII, 1 (1995), p. 155-183 Numdam |  MR 1315354 |  Zbl 0844.35145[3] A. Boulkhemair, “Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators”, Math. Res. Lett. 4 (1997), p. 53-67  MR 1432810 |  Zbl 0905.35103[4] A. Boulkhemair, “$L^2$ estimates for Weyl quantization”, J. Funct. Anal. 165 (1999), p. 173-204 Article |  MR 1696697 |  Zbl 0934.35217[5] R. Coifman & Y. Meyer, Au delà des opérateurs pseudodifférentiels 57, Astérisque, 1978  Zbl 0483.35082[6] C. Fefferman & D. H. Phong, “On positivity of pseudodifferential operators”, Proc. Nat. Acad. Sci. 75 (1978), p. 4673-4674 Article |  MR 507931 |  Zbl 0391.35062[7] L. Hörmander, The analysis of partial differential operators, Springer Verlag, 1985  Zbl 0601.35001[8] N. Lerner & Y. Morimoto, On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators, Preprint, 2005, to appear in the Publications of the Research Institute for Mathematical Sciences (Kyoto University)  MR 2341014[9] N. Lerner & Y. Morimoto, A Wiener algebra for the Fefferman-Phong inequality, in Séminaire EDP, Ecole polytechnique, Exposé no 17, 2005–2006 Cedram |  MR 2276082 |  Zbl 1122.35163[10] J. Sjöstrand, “An algebra of pseudodifferential operators”, Math. Res. Lett. 1,2 (1994), p. 189-192  MR 1266757 |  Zbl 0840.35130[11] D. Tataru, “On the Fefferman-Phong inequality and related problems”, Comm. Partial Differential Equations 27 (2002) no. 11-12, p. 2101-2138 Article |  MR 1944027 |  Zbl 1045.35115 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310