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Phase Space Analysis of Partial Differential Equations

Pseudodifferential systems

speaker: Alberto Parmeggiani (Università di Bologna)

abstract: In these lectures, I plan to give an overview of the almost-positivity of pseudodifferential systems and related spectral problems. I will start by recalling the classical inequalities by G°arding, Melin, H¨ormander and Fefferman-Phong in the scalar case, and by showing methods to detect necessary conditions (both in the scalar and system case), that lead to the concept of localized operator. The spectral properties of the localized operator decide for the possibility of such inequalities. I will then exhibit some classes of systems (constructed starting from the examples of H¨ormander and Brummelhuis) for which the Fefferman-Phong inequality does not hold, and some other classes (systems with symplectic double characteristics) for which the Fefferman-Phong inequality indeed holds true.


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