Elliptic differential operators on Lie groups

AbstractEach strongly elliptic operator in the enveloping Lie algebra associated with a continuous representation of a Lie group generates a holomorphic semigroup S whose action is determined by a universal integral kernel K. We derive pointwise bounds, and Lp-bounds, on K and its derivatives. For example, if the operator is mth order and the group has dimension d then we prove that ¦Kt(g)¦ ⩽ at−dme−b(¦g¦mt)l(m − 1)eωt for all g ϵ G and t > 0, where a, b, and ω are positive constants. The derivative bounds have similar decrease properties as a function over G, the correct singularity as a function of t, and also reflect the joint analyticity of K. One new feature in the analysis is a generalized Nash inequality for the differential structure of the group representation and one application is to Sobolev embeddings for Lie groups