Global Hölder Estimates via Morrey Norms for Hypoelliptic Operators with Drift

Suppose thatX0,X1,…,Xmare left invariant real vector fields on the homogeneous groupGwithX0being homogeneous of degree two andX1,…,Xmhomogeneous of degree one. In the paper we study the hypoelliptic operator with drift of the kindL=∑i,j=1maijXiXj+a0X0,wherea0≠0and(aij)is a constant matrix satisfying the elliptic condition onRm. By proving the boundedness of two integral operators on the Morrey spaces with two weights, we obtain global Hölder estimates forL.

Rough estimates on the scalar heat kernel

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

Evaluation of supertraces for a model operator

This chapter evaluates the supertrace of the heat kernel of a hypoelliptic operator acting over p × g, given a semisimple element γ‎ ∈ G. It begins by introducing a hypoelliptic operator, its heat kernel, and a corresponding supertrace Jᵧ(Y₀ᵗ), if Y₀ᵗ ∈ t(γ‎). Then, by a conjugation of the hypoelliptic operator, the chapter obtains a simpler operator where p × p and t have been decoupled. This new operator splits naturally into a scalar part and a matrix part. Hereafter, the chapter evaluates the trace of the heat kernel of the scalar part, and computes the supertrace of the matrix part of the heat kernel. This chapter concludes with an explicit formula for Jᵧ(Y₀ᵗ).

The hypoelliptic Laplacian on X = G/K

This chapter constructs the hypoelliptic Laplacian ℒbX > 0 acting on the total space of a vector bundle TX ⊕ N ≃ g over the symmetric space X = G/K. The operator ℒbX is obtained using general constructions involving Clifford algebras and Heisenberg algebras, and also the Dirac operator of Kostant. The end result is the elliptic Laplacian 𝓛 X on X as well as the hypoelliptic Laplacian ℒbX, which is a second order hypoelliptic operator acting on X^. Among other things, this chapter gives a key formula relating 𝓛 XℒbX, as well as various formulas involving the operator ℒbX+La.

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.