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On the Fefferman-Phong inequality and related problems. (English) Zbl 1045.35115
The main goal of the paper is to describe a new approach (and an extension to symbols of low regularity) to the Fefferman-Phong inequality and the problem of when a given pseudodifferential operator $$B$$ is controlled by some pseudodifferential operators $$A _{j}$$ in the sense that $$| | B(x ,D)u| | _{L ^{2}}\leq c[ \sum _{j } ^{ }| | A _{j} (x ,D)| | _{L ^{2}} + | | u| | _{ L ^{2}}]$$. (Recall that when $$a (x , \xi )$$ is a non-negative symbol of class $$S ^{2}$$ on $${\mathbb{R}}^{2n}$$, then the Fefferman-Phong inequality says that the associated pseudodifferential operator $$A(x ,D)$$ is semipositive, in the sense that there is a constant $$c>0$$ such that $${\mathcal R} \langle Au ,u \rangle \geq - c| | u| | ^{2}_{L ^{2}}$$.) The basic strategy is to conjugate the operators under consideration with an FBI-transform. This will approximatively diagonalize the principal parts, with a relatively good control of lower order terms. The main technical part in the argument is then a rather subtle lemma on how to write a nonnegative function in $$m$$ variables as a finite sum of squares of functions.

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators
##### Keywords:
Fefferman-Phong inequality; FBI transform
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##### References:
 [1] Jean-Michel Bony, Seminaire: Équations aux Dérivées Partielles, 1998–1999 pp 16– (1999) [2] DOI: 10.1080/03605307808820083 · Zbl 0389.35046 · doi:10.1080/03605307808820083 [3] Jean-Marc Delort, Second Microlocalization and Semilinear Caustics (1992) [4] DOI: 10.1073/pnas.75.10.4673 · Zbl 0391.35062 · doi:10.1073/pnas.75.10.4673 [5] DOI: 10.1090/S0273-0979-1983-15154-6 · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6 [6] Gerald B. Folland, Harmonic Analysis in Phase Space (1989) · Zbl 0682.43001 · doi:10.1515/9781400882427 [7] DOI: 10.1215/S0012-7094-97-08610-5 · Zbl 0879.35059 · doi:10.1215/S0012-7094-97-08610-5 [8] DOI: 10.1007/BF01360085 · Zbl 0090.08101 · doi:10.1007/BF01360085 [9] DOI: 10.2307/1970473 · Zbl 0132.07402 · doi:10.2307/1970473 [10] Lars Hörmander, The Analysis of Linear Partial Differential Operators, I–IV,, 2. ed. (1990) · Zbl 0712.35001 · doi:10.1007/978-3-642-61497-2 [11] DOI: 10.1002/cpa.3160190409 · Zbl 0185.22801 · doi:10.1002/cpa.3160190409 [12] Nicolas Lerner, New Trends in Microlocal Analysis (Tokyo, 1995) pp pp. 23– (1997) · doi:10.1007/978-4-431-68413-8_2 [13] James Ralston, Studies in Partial Differential Equations pp pp. 206– (1982) [14] Johannes Sjöstrand, Astérisque, 95 pp pp. 1– (1982) [15] DOI: 10.1353/ajm.2000.0042 · doi:10.1353/ajm.2000.0042 [16] Daniel Tataru, II. Amer. J. Math. 123 pp 385– (2001) · Zbl 0988.35037 · doi:10.1353/ajm.2001.0021 [17] Michael E. Taylor, Pseudodifferential Operators and Nonlinear PDE (1991) · doi:10.1007/978-1-4612-0431-2
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