A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

Abstract

Abstract. Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Cameron-Martin" Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.

Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups

“…In this section, we review the definitions for infinite-dimensional Heisenberglike groups, which are infinite-dimensional Lie groups based on an abstract Wiener space. Much of the material is this section also appears in [15].…”

“…We revisit the definition of the infinite-dimensional Heisenberg-like groups that were first considered in [11]. Note that since we are interested in subelliptic heat kernel measures on these groups, there are some necessary modifications to the topology as was done in [15]. First we set the following notation which will hold for the rest of the paper.…”

“…is a Brownian motion on G, which is defined explicitly in Proposition 5.1 and Definition 5.2. For all t > 0, let ν t = Law(g t ) denote the heat kernel measure at time t. At this point, let us briefly comment that, as mentioned in [15], it may seem at first glance that the restriction dim(C) < ∞ implies that this subelliptic example is in some sense only finitely many steps from being elliptic. However, this is truly a subelliptic model and the topologies one must deal with in this setting significantly change the standard analysis and introduce several non-trivial complications not present in the elliptic case.…”

“…However, this is truly a subelliptic model and the topologies one must deal with in this setting significantly change the standard analysis and introduce several non-trivial complications not present in the elliptic case. For further discussion, see Section 1.3 of [15].…”

“…In Section 3 we review the definition of the infinite-dimensional Heisenberglike groups first considered in [11]. In the present paper, we choose a topological structure that is better adapted to subellipticity, and in this way our construction parallels [15], where subelliptic heat kernel measure was also studied. In this section, we also review the properties of infinitedimensional Heisenberg-like groups required for the sequel, as well as recalling the Cameron-Martin subgroup and the finite-dimensional projection groups which serve as approximations.…”