Microdifferential operator
≈ 2 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
In mathematics, a microdifferential operator is a linear operator on a cotangent bundle (phase space) that generalizes a differential operator and appears in the framework of microlocal analysis as well as in the Kyoto school of algebraic analysis. The notion was originally introduced by L. Boutet de Monvel and P. Krée as well as by M. Sato, T. Kawai and M. Kashiwara. There is also an approach due to J. Sjöstrand.
In mathematics, a microdifferential operator is a linear operator on a cotangent bundle (phase space) that generalizes a differential operator and appears in the framework of microlocal analysis as well as in the Kyoto school of algebraic analysis. The notion was originally introduced by L. Boutet de Monvel and P. Krée as well as by M. Sato, T. Kawai and M. Kashiwara. There is also an approach due to J. Sjöstrand. P = \sum_{-\infty where each p_j is a holomorphic function on U that is homogeneous of degree j in the second variable. The sheaf \mathcal{E} of microdifferential operators on T^* X is then the subsheaf of \widehat{\mathcal{E}} consisting of those sections satisfying the growth condition on the negative terms; namely, for each compact subset K \subset U , there exists an \epsilon > 0 such that \sum_{j \le 0} \sup_K|p_j| \epsilon^{-j}/(-j)! ===Notes=== ===Works=== Aoki, T., Calcul exponentiel des opérateurs microdifférentiels d'ordre infini, I, Ann. Inst. Fourier, Grenoble, 33–4 (1983), 227–250. Boutet De Monvel, Louis ; Krée, Paul, Pseudo-differential operators and Gevrey classes, Annales de l'Institut Fourier, Volume 17 (1967) no. 1, pp. 295-323 M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, in: Lecture Notes in Math. 287, Springer, 1973, 265–529. Sjöstrand, Johannes. Singularités analytiques microlocales, dans Singularités analytiques microlocales - équation de Schrödinger et propagation des singularités..., Astérisque, no. 95 (1982), pp. iii-166. https://www.numdam.org/item/AST_1982__95__R3_0/
Text derived from https://en.wikipedia.org/w/index.php?oldid=1322055642 under CC BY-SA 4.0; modifications have been made.