Kernel calculus and extension of contact transformation to \(D\)-modules. (English) Zbl 0872.32007

Bony, J.-M. (ed.) et al., New trends in microlocal analysis. Tokyo: Springer. 179-190 (1997).
From the introduction: “There is an important literature dealing with integral transformations. In our papers [A. D’Agnolo and P. Schapira, J. Funct. Anal. 139, No. 2, 349-382, Art. No. 0089 (1996); Duke Math. J. 84, No. 2, 453-496 (1996) and C. R. Acad. Sci., Paris, Sér. I 319, No. 5, 461-466 (1994; Zbl 0827.32008)], we proposed a general framework to the study of such transformations in the language of sheaves and \(D\)-modules. In particular, we showed that there are two natural adjunction formulas.”
“Here, we shall first recall the four above mentioned adjunction formulas, and then concentrate our study on the \(D\)-module theoretical transform. Given two complex manifolds \(X\) and \(Y\) and a \(D\)-module kernel which defines a quantized contact transformation on an open subset of \(T^{\ast} (X \times Y)\), our main result gives a geometrical condition to extend it as an isomorphism of locally free \(D\)-modules of rank one. This improves our previous result of the third paper, cited above”.
For the entire collection see [Zbl 0859.00023].


32C38 Sheaves of differential operators and their modules, \(D\)-modules
58J35 Heat and other parabolic equation methods for PDEs on manifolds


Zbl 0827.32008