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

Full-text PDF Free Access

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.

**[1]**M. Breuer and H. Cordes,*On Banach algebras with -symbol*, J. Math. Mech.**13**(1964), 313-324. MR**28**#2456. MR**0159239 (28:2456)****[2]**H. Cordes and E. Herman,*Gelfand theory of pseudo differential operators*, Amer. J. Math.**90**(1968), 681-717. MR**0454743 (56:12991)****[3]**J. Dixmier,*Les -algèbres et leurs répresentations*, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR**30**#1404. MR**0171173 (30:1404)****[4]**L. Hörmander,*Linear partial differential operators*, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR**28**#4221.**[5]**-,*Pseudo-differential operators and hypoelliptic equations*, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp. 138-183. MR**0383152 (52:4033)****[6]**-,*On interior regularity of the solutions of partial differential equations*, Comm. Pure Appl. Math.**11**(1958), 197-218. MR**21**#5064. MR**0106330 (21:5064)****[7]**B. Malgrange,*Sur une classe d'opérateurs différentiels hypoelliptiques*, Bull. Soc. Math. France**85**(1957), 283-306. MR**21**#6063. MR**0106329 (21:5063)****[8]**E. Nelson and W. F. Stinespring,*Representation of elliptic operators in an enveloping algebra*, Amer. J. Math.**81**(1959), 547-560. MR**22**#907. MR**0110024 (22:907)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47G05,
35H05

Retrieve articles in all journals with MSC: 47G05, 35H05

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