Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups

Abstract

Let (χ, G, U) be a continuous representation of a Lie groupG by bounded operatorsg →U(g) on the Banach space χ and let (χ,$\mathfrak{g}$,dU) denote the representation of the Lie algebra$\mathfrak{g}$ obtained by differentiation. Ifa ₁, ...,a _d′ is a Lie algebra basis of$\mathfrak{g}$,A _i=dU(a _i) and$A^\alpha = A_{i_1 } ...A_{i_k }$ whenever α=(i ₁, ...,i _k) we consider the operators

$$H = \mathop \sum \limits_{\alpha ;|\alpha | \leqslant 2n} c_\alpha A^\alpha $$

where thec _α are complex coefficients satisfying a subcoercivity condition. This condition is such that the class of operators considered encompasses all the standard second-order subelliptic operators with real coefficients, all operators of the form$\sum _{i = 1}^{d'} \lambda _i ( - A_i^2 )^n$ with Re λ _i >0 together with operators of the form

$$H = ( - 1)^n \mathop \sum \limits_{\alpha ;|\alpha | = n} \mathop \sum \limits_{\beta ;|\beta | = n} c_{\alpha ,\beta } A^{\alpha _* } A^\beta $$

where α_*=(i _k, ...,i ₁) if α=(i ₁, ...,i _k) and the real part of the matrix (c _{α β}) is strictly positive. In case the Lie algebra$\mathfrak{g}$ is free of stepr, wherer is the rank of the algebraic basisa ₁, ...,a _d′,G is connected andU is the left regular representation inG we prove that the closure$\overline H$ ofH generates a holomorphic semigroupS. Moreover, the semigroupS has a smooth kernel and we derive bounds on the kernel and all its derivatives. This will be a key ingredient for the paper [13] in which the above results will be extended to general groups and representations.