Subunit balls for symbols of pseudodifferential operators. (English) Zbl 0940.35214

Parmeggiani, Alberto

Adv. Math. 131, No. 2, 357-452 (1997).

The author defines subunit balls for non-negative symbols in the class \(S^2(1\times M)\) of subelliptic pseudodifferential operators on \({\mathbb R}^n\), extending in phase-space, the definition given by A. Nagel, E. M. Stein and S. Wainger [Acta Math. 155, 103-147 (1985; Zbl 0578.32044)] in the differential operator case. With the use of microlocal methods introduced by C. L. Fefferman and D. H. Phong [Commun. Pure Appl. Math. 34, 285-331 (1981; Zbl 0458.35099)], it is proved that these balls can be straightened, by means of a canonical transformation, to contain and be contained in boxes of certain sizes, which are given in terms of the size of the symbol. For \(n=1\) and for classes of, for instance, \(n=2\), one obtains a complete understanding of the non-Euclidean balls, which are comparable to rectangular boxes after a suitable canonical transformation. However, for \(n=2\), an example is given which exhibits a new phenomenon, called stratification, with no analogue for the differential operator case. Moreover, it is conjectured that, in the general case, the ball \(B_p((x,\xi),\rho)\) with radius \(\rho\) comparable to one of a bounded number of critical radii \(\rho_1,\ldots,\rho_N\), with \(N\) bounded a priori, no longer looks like a box, and \(B_p((x,\xi),4\rho)\) is very large compared to \(B_p((x,\xi),\rho/4)\).

Reviewer: R.Vaillancourt (Ottawa)

MSC:

35S05	Pseudodifferential operators as generalizations of partial differential operators
47G30	Pseudodifferential operators
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35H20	Subelliptic equations

Keywords:

subelliptic pseudodifferential operators; subunit balls; microlocalization; stratification; straightening

Citations:

Zbl 0578.32044; Zbl 0458.35099

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