Bounds on the Segal-Bargmann transform ofL p functions

Abstract

This article gives necessary conditions and slightly stronger sufficient conditions for a holo- morphic function to be the Segal-Bargmann transform of a function in L p (ll~ d, p), where p is a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be "tuned" to give the best estimates for a given value of p. This article also gives a single necessary-and- sufficient condition for a holomorphic function to be the transform of a function f such that any derivative of f multiplied by any polynomial is in L P (~d, p). Finally, l give some weaker but dimension-independent conditions. Gaussian measure p (x) dx. We will let LP(R a, p) stand for LP(~ a, p (x) dx). Note that p (x) has an entire analytic continuation to C d. Now consider f e LP(R d, p), with 1 < p < 00. Define the Segal-Bargmann transform Sf of f by Sf (z) = .f~a p (z - x) f (x) dx ,