Holomorphic functions and the Itô chaos

Driver, Bruce K.

doi:10.2969/jmsj/06741449

Cited by 1 publication

(3 citation statements)

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“…It is well known that the evaluation map F → F (z) on HL 2 (K C , µ s,τ ) is a bounded linear functional; this is due to the ubiquitous pointwise L 2 estimates in this holomorphic space (cf. [7,21]). We claim that we can actually find locally uniform bounds on this functional.…”

Section: Complex Vector Fields and Commutation Relationsmentioning

confidence: 99%

“…To establish the bound in (5.13), we observe that the norm of the pointwise evaluation functional can be estimated in terms of lower bounds on the density µ s,τ . For example, [7,Theorem 3.6] shows (in our context) that, for any precompact neighborhood V of the identity e, there is a constant C(V ) so that, for all holomorphic F and z ∈ K C ,…”

Section: Complex Vector Fields and Commutation Relationsmentioning

confidence: 99%

“…Furthermore, the pointwise evaluation map F → F (z) is continuous for each z ∈ M , and the norm of this functional is locally bounded as a function of z. (See, for example, Theorem 3.2 and Corollary 3.3 in [7] or Theorem 2.2 in [21]. )…”

Section: Introductionmentioning

confidence: 99%

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The complex-time Segal–Bargmann transform

Driver

Hall

Kemp

2020

Journal of Functional Analysis

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We introduce a new form of the Segal-Bargmann transform for a Lie group K of compact type. We show that the heat kernel (ρt(x)) t>0,x∈K has a space-time analytic continuation to a holomorphic function (ρ C (τ, z)) Re τ >0,z∈K C , where K C is the complexification of K. The new transform is defined by the integralIf s > 0 and τ ∈ D(s, s) (the disk of radius s centered at s), this integral defines a holomorphic function on K C for each f ∈ L 2 (K, ρs). We construct a heat kernel density µs,τ on K C such that, for all s, τ as above, Bs,τ := Bτ | L 2 (K,ρs) is an isometric isomorphism from L 2 (K, ρs) onto the space of holomorphic functions in L 2 (K C , µs,τ ). When τ = t = s, the transform Bt,t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ = t ∈ (0, 2s), the transform Bs,t coincides with the one introduced by the first two authors.

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Section: Complex Vector Fields and Commutation Relationsmentioning

confidence: 99%

Section: Complex Vector Fields and Commutation Relationsmentioning

confidence: 99%