Subcoercivity and subelliptic operators on Lie groups I

Abstract

Let (χ, G, U) be a continuous representation of a Lie group G by bounded operators g →U(g) on the Banach space χ and let (χ, {Mathematical expression}, dU) denote the representation of the Lie algebra {Mathematical expression} obtained by differentiation. If a1, ..., ad′ is a Lie algebra basis of {Mathematical expression}, Ai=dU(ai) and {Mathematical expression} whenever α=(i1, ..., ik) we consider the operators {Mathematical expression} where the cα 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 {Mathematical expression} with Re λi>0 together with operators of the form {Mathematical expression} where α*=(ik, ..., i1) if α=(i1, ..., ik) and the real part of the matrix (cα β) is strictly positive. In case the Lie algebra {Mathematical expression} is free of step r, where r is the rank of the algebraic basis a1, ..., ad′, G is connected and U is the left regular representation in G we prove that the closure {Mathematical expression} of H generates a holomorphic semigroup S. Moreover, the semigroup S 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. Show more