Isomorphisms of pseudo-differential operators in the SG-calculus
Venue: Alessandro Contini - Aula 3 (Dipartimento di Matematica, Università di Torino)
The algebraic relationship between (classical) pseudo-differential ($\Psi$DO) and Fourier Integral Operators (FIO) (beyond the module structure) is expressed by well-known results by Egorov and Duistermaat-Singer. Namely, conjugating a $\Psi$DO with an (elliptic) FIO returns a $\Psi$DO, whose principal symbol is given by pull-back along the underlying symplectomorphism of the original one. Viceversa, any isomorphism of algebras of $\Psi$DO preserving the order of all operators must necessarily be of the above form. In this talk, based on the contents of my PhD thesis, we will explore the corresponding question for the algebra of $SG$-$\Psi$DOs. This is a well-studied class of operators, obtained by introducing a second, independent, order, measuring the growth of the symbols as $|x|\rightarrow\infty$. FIOs in this classes have also been introduced (with various degrees of generality and geometric content), and the Egorov theorem has been shown to hold true. We shall therefore address the question of a result á la Duistermaat-Singer in this framework, especially looking at some proof ideas and problems that we encounter. Time permitting, we shall also discuss the notion of symplectomorphism that appears naturally here (in fact, a contact morphism between manifolds with corners).
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