Gevrey hypoellipticity for non-subelliptic operators (Q612258)

The operator in \(\mathbb{R}^3_{x,y,t}\) \[ E_{m,k}= L_m\overline L_m+\overline L_m|z|^{2k} L_m \] with \[ L_m= \partial/\partial z- i\overline z|z^{2(m-1)}\partial/\partial t,\quad z= x+ iy, \] was proved to be hypoelliptic with loss of \((k-1)/m\) derivatives by \textit{J. J. Kohn} [Ann. Math. (2) 162, No. 2, 943--986 (2005; Zbl 1107.35044)]. Here the authors consider the operator in \(\mathbb{R}^4_{x,y,t,v}\) \[ E_{m,k,p}= E_{m,k}+ |z|^{2p-1)}\partial^2/\partial v^2. \] This operator is proved to be hypoelliptic and \(s\)-Gevrey hypoelliptic for any \(s\geq 2m/(p- k)\).