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

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 C3,1 nonnegative functions as a sum of squares of C1,1 functions with sharp estimates. In particular, we prove that a C3,1 nonnegative function a can be written as a finite sum Σ b2j, where each bj is C1,1, but also where each function b2j is C3,1. A key point in our proof is to give some bounds on (bj' bj'')' and on (bj' bj'')''.