The analysis of the hypoelliptic Laplacian
This chapter constructs a functional analytic machinery that is adapted to the analysis of the hypoelliptic Laplacian ℒA,bX. The analysis of the hypoelliptic Laplacian essentially consists in the construction of Sobolev spaces on which the operators ℒA,bX act as unbounded operators, and in the proof of regularizing properties of their resolvents and of their heat operators. The heat operators are shown to be given by smooth kernels. To make the analysis easier, the chapter first works with a scalar hypoelliptic operator AbX. This operator does not contain a quartic term. The results on AbX then easily extend to a scalar operator AbX acting over X with circumflex, which also does not contain the quartic term. A scalar operator AbX on ̂X containing the quartic term is introduced. Finally, the chapter extends the analysis to the operator ℒA,bX.