### On the Fefferman-Phong inequality

Abdesslam Boulkhemair (2008)

Annales de l’institut Fourier

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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 $\frac{n}{2}+4+\u03f5$ improving thus the bound $2n+4+\u03f5$ 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 $n+4+\u03f5$; 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, $4\le \left|\alpha \right|+\left|\beta \right|\le n+4+\u03f5$. To obtain such results and others, we first prove an abstract result which...