Some Remarks on Hypoelliptic Operators which are not Micro-hypoelliptic
Yoshinori Morimoto, Tatsushi Morioka · Publications of the Research Institute for Mathematical Sciences · 1992
In this note we give an example of hypoelliptic operators which are not micro-hypoelliptic. Non-micro-hypoellipticiy of the example arises from the oscillation of the coefficient with a zero of infinite order. Let us consider the following semi-elliptic operator with infinite degeneracy: (1.1) L = a(x, .y, Dx) + g(x)b(x, y, Dy) in R = R'x l X R?,. Here g(jc) E C and satisfies (A.I) g(x)>0 for *^0 and 5^(0) = 0 for any /3. Here a(x, y, Dx} and b(x, y, DY) are differential operators with C x coefficients of order 2€ and 2m. We assume that a(x, y, DY) and b(x, y, Dv) are strongly elliptic with respect to x and y, respectively, that is, for Q, C2>0 (A.2) Re a(x, y, ?) > Q | | and (A.3) Re &(*, y, ?7)>C2|?7| 2m hold if | §| and | ry | are sufficiently large. In [3] the one of authors (T.M.) proved that the operator L is hypoelliptic, i.e. (1.2) sing supp u = sing supp Lu for w E 2)'.