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Subunit Balls for Symbols of Pseudodifferential Operators

✍ Scribed by Alberto Parmeggiani

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 880 KB
Volume: 131
Category: Article
ISSN: 0001-8708
DOI: 10.1006/aima.1997.1672

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✦ Synopsis

In this work we shall study a definition of subunit ball for non-negative symbols of sub-elliptic pseudodifferential operators, extending in phase-space the one given by Stein, Nagel, and Wainger in the differential-operator case. Using microlocal methods introduced by Fefferman and Phong, we prove that these balls can be straightened, by means of a canonical transformation, to contain and be contained in boxes of certain sizes, which we give in terms of the size of the symbol. After microlocalizing the symbol, in Section 3 we define classes of subunit symbols and study some of their basic properties. Then we define the subunit ball. In the last section the main structure theorems, in the (n+n)-dimensional elliptic case and in the (1+1)-and (2+2)-dimensional nonelliptic nondegenerate cases are stated and proved.

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