On a class of linear differential operators of infinite order with finite index

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書誌事項

公開日
1986-11
権利情報
  • https://www.elsevier.com/tdm/userlicense/1.0/
  • https://www.elsevier.com/open-access/userlicense/1.0/
DOI
  • 10.1016/0001-8708(86)90098-8
公開者
Elsevier BV

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説明

The recent progress of the theory of microdifferential operators of infinite order [l-4] and their usefulness in several branches of analysis [IS, 11, 16-241 have enhanced the interest in such operators. However, our knowledge on such operators is still quite limited, particularly compared with our knowledge on operators of finite order. For example, we do not know much about such a basic problem as characterizing a linear differen- tial operator of infinite order which is of closed range as a map from germs of holomorphic functions to germs of holomorphic functions. To the best of our knowledge, Ishimura’s recent work [7] is the only result that has some general applicability, although it is still far from the final goal. The purpose of this paper is to prove some theorems which we hope to contribute toward the progress of the study of such operators. To be con- crete, we present some conditions on a linear differential operator P which guarantees the finiteness of the dimensions of the kernel and the cokernel of the map P: Ccn,o -+ O,,,,, where Ocn,O denotes germ at origin of sheaf 0,. of holomorphic functions on C:“. In particular, under those con- ditions the map has finite index, and it is of closed range (Theorem 4 and Theorem 5). Our results are a natural generalization of Bony and Schapira [S, Thtoreme 5.11 to the infinite-order case. It is also quite akin to Kashiwara, Kawai, and Sjostrand [12] in its philosophy in the sense that some ellipticity condition for a suitably chosen tangential system lies 155

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