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
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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 dimensions is bounded above by . Our method relies on some refinements of the Wick calculus, which is closely linked to Gabor wavelets. Also we use a decomposition of nonnegative functions as a sum of squares of functions with sharp estimates. In particular, we prove that a nonnegative function can be written as a finite sum , where each is , but also where each function is . A key point in our proof is to give some bounds on and on .
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