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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
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