## 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].

### MSC:

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

### Keywords:

$$D$$-module; quantized contact transformation

Zbl 0827.32008