Analytic Pseudodifferential Operators
“Generalized functions,” playing the role vis-à-vis analytic functions that distributions play vis-à-vis C ∞ functions, do exist: They are the hyper-functions of M. Sato. The analogues of pseudodifferential (and even hyperdifferential) operators can be made to act on them, and many results in this book have hyperfunction parallels. On this vast subject we refer the reader to Sato, Kawai, and Kashiwara . In general, that is, in the absence of precise information, such operators transform hyperfunctions, and among them distributions, into hyperfunctions. As a consequence there may still be some justification for a theory of pseudodifferential operators that transform distributions into distributions while preserving their analyticity. This chapter is devoted to the definition and study of standard pseudodifferential operators (in the sense of Chapter I) having this property.
KeywordsOpen Subset Open Neighborhood Pseudodifferential Operator Cutoff Function Open Cone
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