Gelfand theory of pseudo differential operators and hypoelliptic operators
Author:
Michael E. Taylor
Journal:
Trans. Amer. Math. Soc. 153 (1971), 495-510
MSC:
Primary 47G05; Secondary 35H05
DOI:
https://doi.org/10.1090/S0002-9947-1971-0415430-8
MathSciNet review:
0415430
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Abstract | References | Similar Articles | Additional Information
Abstract: This paper investigates an algebra of pseudo differential operators generated by functions
such that
as
, if
, and by operators
where
, and
is hypoelliptic. It is proved that such an algebra has compact commutants, and the maximal ideal space of the commutative
algebra
is investigated, where
consists of the elements of
which are compact. This gives a necessary and sufficient condition for a differential operator
to be Fredholm. (Here and in the rest of this paragraph we assume that the coefficients of all operators under consideration satisfy the conditions given on
in the first sentence.) It is also proved that if
is a formally selfadjoint operator on
which has the same strength as
uniformly on
, then
is selfadjoint, with domain
, and semibounded, if
. From this a Gårding type inequality for uniformly strongly formally hypoelliptic operators and a global regularity theorem for uniformly formally hypoelliptic operators are derived. The familiar local regularity theorem is also rederived.
It is also proved that a hypoelliptic operator of constant strength is formally hypoelliptic, in the sense that for any
, the constant coefficients operator
is hypoelliptic.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1971-0415430-8
Keywords:
Pseudo differential operator,
hypoelliptic operator,
Fredholm operator
Article copyright:
© Copyright 1971
American Mathematical Society