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Hypoellipticity and Local Solvability in Gevrey Classes
Author(s) -
Albanese Angela A.,
Corli Andrea,
Rodino Luigi
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200207)242:1<5::aid-mana5>3.0.co;2-e
Subject(s) - mathematics , hypoelliptic operator , pure mathematics , operator (biology) , differential operator , class (philosophy) , mathematical analysis , semi elliptic operator , biochemistry , chemistry , repressor , artificial intelligence , computer science , transcription factor , gene
Let P be a linear partial differential operator with coefficients in the Gevrey class G s . We prove first that if P is s ‐hypoelliptic then its transposed operator t P is s ‐locally solvable, thus extending to the Gevrey classes the well‐known analogous result in the C ∞ class. We prove also that if P is s ‐hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s ‐hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.