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 φ(x) and the sum of squares of first order operators X₁,…,X_r. These conditions are related to the way in which φ(x) changes its sign, and the rank of the Lie algebra generated by φX₁,…,φX_r and X₀ where X₀ 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].