# On the Fefferman–Phong Inequality and a Wiener-type Algebra of Pseudodifferential Operators

### Nicolas Lerner

Institut Mathématiques de Jussieu, Paris, France### Yoshinori Morimoto

Kyoto University, Japan

## Abstract

We provide an extension of the Fefferman–Phong inequality to nonnegative symbols whose fourth derivative belongs to a Wiener-type algebra of pseudodifferential operators introduced by J. Sjöstrand. As a byproduct, we obtain that the number of derivatives needed to get the classical Fefferman–Phong inequality in $d$ dimensions is bounded above by $2d+4+ϵ$. Our method relies on some reﬁnements of the Wick calculus, which is closely linked to Gabor wavelets. Also we use a decomposition of $C_{3,1}$ nonnegative functions as a sum of squares of $C_{1,1}$ functions with sharp estimates. In particular, we prove that a $C_{3,1}$ nonnegative function $a$ can be written as a ﬁnite sum $∑b_{j}$, where each $b_{j}$ is $C_{1,1}$, but also where each function $b_{j}$ is $C_{3,1}$. A key point in our proof is to give some bounds on $(_{b}j_{′}b_{j})_{′}$ and on $(b_{j}b_{j})_{′′}$.

## Cite this article

Nicolas Lerner, Yoshinori Morimoto, On the Fefferman–Phong Inequality and a Wiener-type Algebra of Pseudodifferential Operators. Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, pp. 329–371

DOI 10.2977/PRIMS/1201011785