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On the Fefferman-Phong inequality. (Sur l’inégalité de Fefferman-Phong.) (French) Zbl 1086.35529
The author investigates the validity for a pseudo-differential operator $$A$$ of the estimates $$(Af,f)\geq-C\|f\|_{L^2}^2$$. The operator $$A$$ is assumed to be selfadjoint, that is, using the Weyl quantization $$A=a^w(x,D)$$, the symbol $$a(x,\xi)$$ is assumed to be nonnegative. Several results are presented, extending the Gårding and Fefferman-Phong inequalities in the frame of the Weyl-Hörmander calculus; in particular the author proves that if $$|D_x^\alpha D_\xi^\beta a(x,\xi)|\leq C$$ for $$|\alpha+\beta|\geq4$$, then the above estimates are satisfied.

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators
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