Analytic-gevrey hypoellipticity for a class of pseudo-differential operators with multiple characteristics

Let $P(x, D)$ be a partial differential operator with principal symbol $p_m(x, \xi)=q_{m-l}(x, \xi)a_1(x, \xi)^l$, $l\ge 1$, where $q_{m-l}(x,\xi)$ is elliptic of order $m-l$, $a_1(x, \xi)$ is complex-valued and of principal type. For such operators the authors prove Theorem 1.1: If $\roman{Im}\,a_1(x, \xi)$ has a zero of fixed finite odd order $h>0$ on every bicharacteristic of $\roman{Re}\,a_1(x, \xi)$ near $(x_0, \xi_0)$ and (i) the change of sign is from $-$ to $+$, then $P$ is $s$-micro-hypoelliptic at $(x_0, \xi_0)$ for $1\le s<l/(l-1)$. (ii) If the change of sign of $\roman{Re}\,a_1(x,\xi)$ is from $+$ to $-$ then $P$ is not $s$-micro-hypoelliptic at $(x_0, \xi_0)$ for $1\le s<l/(l-1)$. Next, the authors give an example of analytic hypoellipticity but not $C^\infty$-hypoellipticity, namely, they prove Theorem 3.1: For $P=\sum_{\alpha +k\beta \le l}c_{\alpha \beta} x^{\gamma(\alpha ,\beta)}(D_x+ix^k D_y)^\alpha D_y^\beta$, under certain conditions, analytic hypoellipticity of $P$ does not imply its $C^\infty$-hypoellipticity.