Hypoellipticity of a second order operator with a principal symbol changing sign across a smooth hypersurface

Abstract

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 $\phi \left(x\right)$ and the sum of squares of first order operators $X_1,\dots ,X_r$. These conditions are related to the way in which $\phi \left(x\right)$ changes its sign, and the rank of the Lie algebra generated by $\phi X_1,\dots ,\phi X_r$ and $X_0$ where $X_0$ is the first order term of the operator. Our result is an extension of that of [4], and it includes some cases not treated in [1], [5] and [8].