The uncertainty principle and hypoelliptic operators
Let P be a differential operator of second order with coefficients in \(C^{\infty}(R^ n)\), that is, \[ (1)\quad P=\sum_{j,k}a_{jk}(x)D_ jD_ k+\sum_{j}ib_ j(x)D_ j+c(x),\quad D_ j=-i\partial_{x_ j}. \] We assume that \[ (2)\quad a_{jk}\quad and\quad b_ j\quad are\quad real\quad valued,\quad \sum a_{jk}(x)\xi_ j\xi_ k\geq 0\quad for\quad any\quad (x,\xi)\in R^{2n}. \] Let log \(\Lambda\) be a pseudodifferential operator with symbol \(\log <\xi >\). As a criterion for P to be hypoelliptic in \(R^ n\) the author has given the following (see also the preceding review) Theorem A. If for any \(\epsilon >0\) and any compact set K of \(R^ n\) there exists a constant \(C_{\epsilon,K}\) such that \[ (3)\quad \| (\log \Lambda)u\|^ 2\leq \epsilon Re(Pu,u)+C_{\epsilon,K}\| u\|^ 2,\quad u\in C^{\infty}_ 0(K), \] then P is hypoelliptic in \(R^ n\). Furthermore, we have \[ (4)\quad WF P\upsilon =WF \upsilon \quad for\quad \upsilon \in {\mathcal D}'(R^ n). \] Though more general criteria than Theorem A were given by the author [Osaka J. Math. 24, 651- 675 (1987; Zbl 0644.35023) and Semin., Equations Deriv. Partielles 1986- 1987, Exp. No.15, 12 p. (1987; Zbl 0644.35022)], in the present paper we restrict our consideration to the application of this theorem to some class of degenerate elliptic operators of second order.
scientific article; zbMATH DE number 4075639
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