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On the Fefferman-Phong inequality. (English) Zbl 1145.35099
Summary: 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 $$\frac{n}{ 2}+4+\varepsilon$$ improving thus the bound $$2n+4+\varepsilon$$ 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 $$n+4+\varepsilon$$; 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, $$4\leq |\alpha |+|\beta |\leq n+4+\varepsilon$$. 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 $$L^2$$.
##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators 58J40 Pseudodifferential and Fourier integral operators on manifolds
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##### References:
 [1] Bony, J.-M., Sur l’inégalité de Fefferman-phong, Séminaire EDP, (19981999), Ecole polytechnique · Zbl 1086.35529 [2] Boulkhemair, A.$$, L^2$$ estimates for pseudodifferential operators, Ann. Sc. Norm. Sup. Pisa, IV, XXII, 1, 155-183, (1995) · Zbl 0844.35145 [3] Boulkhemair, A., Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., 4, 53-67, (1997) · Zbl 0905.35103 [4] Boulkhemair, A.$$, L^2$$ estimates for Weyl quantization, J. Funct. Anal., 165, 173-204, (1999) · Zbl 0934.35217 [5] Coifman, R.; Meyer, Y., Au delà des opérateurs pseudodifférentiels, 57, (1978), Astérisque · Zbl 0483.35082 [6] Fefferman, C.; Phong, D. H., On positivity of pseudodifferential operators, Proc. Nat. Acad. Sci., 75, 4673-4674, (1978) · Zbl 0391.35062 [7] Hörmander, L., The analysis of partial differential operators, (1985), Springer Verlag · Zbl 0601.35001 [8] Lerner, N.; Morimoto, Y., On the Fefferman-Phong inequality and a Wiener type algebra of pseudodifferential operators, (2005), Preprint [9] Lerner, N.; Morimoto, Y., A Wiener algebra for the Fefferman-phong inequality, Séminaire EDP, (20052006), Ecole polytechnique · Zbl 1122.35163 [10] Sjöstrand, J., An algebra of pseudodifferential operators, Math. Res. Lett., 1,2, 189-192, (1994) · Zbl 0840.35130 [11] Tataru, D., On the Fefferman-phong inequality and related problems, Comm. Partial Differential Equations, 27, 11-12, 2101-2138, (2002) · Zbl 1045.35115
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