# 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 2_d_ + 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 Σ b_2_j, where each bj is C_1,1, but also where each function b_2_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*'')''.