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

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 $\sum b_j^2$, where each $b_j$ is $C^{1,1}$, but also where each function $b_j^2$ is $C^{3,1}$. A key point in our proof is to give some bounds on ${(}_b{j}^{\prime }{b}_j^{\prime \prime }{)}^{\prime }$ and on $(b_j^{\prime }{b}_j^{\prime \prime }{)}^{\prime \prime }$.