Gelfand theory of pseudo differential operators and hypoelliptic operators

This paper investigates an algebra $\mathfrak{A}$ of pseudo differential operators generated by functions $a(x) \in {C^\infty }({R^n}) \cap {L^\infty }({R^n})$ such that ${D^\alpha }a(x) \to 0$ as $\vert x\vert \to \infty$, if $\vert\alpha \vert \geqq 1$, and by operators $q(D){Q^{ - 1/2}}$ where $q(D) < P(D),Q = I + P{(D)^ \ast }P(D)$, and $P(D)$ is hypoelliptic. It is proved that such an algebra has compact commutants, and the maximal ideal space of the commutative ${C^ \ast }$ algebra $\mathfrak{A}/J$ is investigated, where $J$ consists of the elements of $\mathfrak{A}$ which are compact. This gives a necessary and sufficient condition for a differential operator $q(x,D):{\mathfrak{B}_2}_{,\tilde P} \to {L^2}$ 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 $a(x)$ in the first sentence.) It is also proved that if $p(x,D)$ is a formally selfadjoint operator on ${R^n}$ which has the same strength as $P(D)$ uniformly on ${R^n}$, then $p(x,D)$ is selfadjoint, with domain ${\mathfrak{B}_{2,\tilde P}}({R^n})$, and semibounded, if $n \geqq 2$. 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 $p(x,D)$ of constant strength is formally hypoelliptic, in the sense that for any ${x_0}$, the constant coefficients operator $p({x_0},D)$ is hypoelliptic.