We give sufficient conditions for hypoellipticity of a second order operator with real-valued infinitely differentiable coefficients whose principal part is the product of a real-valued infinitely differentiable function and the sum of squares of first order operators . These conditions are related to the way in which changes its sign, and the rank of the Lie algebra generated by and where is the first order term of the operator. Our result is an extension of that of , and it includes some cases not treated in ,  and .
Toyohiro AKAMATSU. "Hypoellipticity of a second order operator with a principal symbol changing sign across a smooth hypersurface." J. Math. Soc. Japan 58 (4) 1037 - 1077, October, 2006. https://doi.org/10.2969/jmsj/1179759537