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

We provide an extension of the Fefferman-Phong inequality to nonnegative sym- bols whose fourth derivative belongs to a Wiener-type algebra of pseudodifferential operators introduced by J. Sjostrand. 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+ . 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