Abstract
Maslov’s canonical operator on punctured Lagrangian manifolds provides a solution to the Cauchy problem with initial data concentrated near a point or a submanifold of positive codimension for equations and wave-type systems for which the roots of the characteristic equation have singularities such as nonsmoothness and/or change of multiplicities at zero values of momenta. The theory of the canonical operator on punctured Lagrangian manifolds was constructed in the article [1] by S. Yu. Dobrokhotov, A. I. Shafarevich, and the author, in which, however, the formula for commutation of the canonical operator with pseudodifferential operators was not given. This formula is proved in the present article; moreover, the construction of the canonical operator on punctured Lagrangian manifolds is presented in an equivalent, more convenient form. We restrict ourselves to the local theory (the precanonical operator, or the operator in a separate chart of the Lagrangian manifold corresponding to some nondegenerate phase function), since the transition to the global construction does not contain anything new compared to the standard case.
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Notes
In what follows, this property is called pseudohomogeneity, so as not to be confused with asymptotic homogeneity at infinity.
Not to be confused with blow-up in algebraic geometry.
If we choose a different section \(\widetilde s\), then the corresponding embedding \(W\subset\widehat W\) will be different as well; on each orbit \(\pi_W^{-1}(\omega)\), it differs from the original embedding by the action of the element \(\lambda\in\mathbb{R}_+\) such that \(\widetilde s(\omega)=\lambda\cdot s(\omega)\).
An alternative definition of a semiconical set that does not use the ambient manifold \(W\) can, say, be stated as follows: \(V\) is a topological space equipped with a local action of the group \(\mathbb{R}_+\), and for any \(v\in V\) the set of elements \(\lambda\in\mathbb{R}_+\) for which the action \(\lambda\cdot v\) is defined contains the half-interval \((0,1]\).
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Acknowledgments
The author wishes to express gratitude to S. Yu. Dobrokhotov and A. I. Shafarevich for numerous helpful discussions.
Funding
The present work was supported by the Ministry of Science and Higher Education within the framework of the Russian State Assignment under contract No. AAAA-A20-120011690131-7.
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 733–751 https://doi.org/10.4213/mzm13672.
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Nazaikinskii, V.E. Canonical Operator on Punctured Lagrangian Manifolds and Commutation with Pseudodifferential Operators: Local Theory. Math Notes 112, 709–725 (2022). https://doi.org/10.1134/S0001434622110086
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DOI: https://doi.org/10.1134/S0001434622110086