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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 \(L^2\).
35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
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