On the Fefferman–Phong Inequality and a Wiener-type Algebra of Pseudodifferential Operators
Nicolas Lerner
Institut Mathématiques de Jussieu, Paris, FranceYoshinori 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'')''.
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