Germ hypoellipticity and loss of derivatives

We prove hypoellipticity in the sense of germs for the operator

$$\mathcal {P}= L_{q}\overline {L}_{q} + \overline {L}_{q}t^{2k}L_{q} +Q^{2},$$

where

$$L_{q}=D_{t}+it^{q-1}\sqrt {-\Delta _{x}}$$ $$\quad \text {and}\quad Q = x_{1}D_{2}-x_{2}D_{1},$$

even though it fails to be hypoelliptic in the strong sense. The primary tool is an a priori estimate.