[Paper Review] Bounds on the Segal-Bargmann transform of L^p functions

Brian C. Hall|ArXiv.org|Mar 1, 2001

Mathematical Analysis and Transform Methods15 references4 citations

TL;DR

This paper establishes sharp bounds characterizing the image of $ L^p(\mathbb{R}^d, \rho) $ under the Segal-Bargmann transform, where $ \rho $ is the Gaussian measure. Using tuned inversion formulas and Hölder's inequality, it provides nearly optimal necessary and sufficient conditions for a holomorphic function to be such a transform, with dimension-independent results derived from Nelson's hypercontractivity.

ABSTRACT

This paper gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function in L^p(R^d) with respect to 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.

Motivation & Objective

• To precisely characterize the set of holomorphic functions on $ \mathbb{C}^d $ that arise as the Segal-Bargmann transform of $ L^p(\mathbb{R}^d, \rho) $ functions for $ 1 < p < \infty $.
• To derive necessary and slightly stronger sufficient conditions for a holomorphic function to be in the image of $ L^p(\mathbb{R}^d, \rho) $ under the Segal-Bargmann transform.
• To extend these results to the $ L^p $-Schwartz space, where all derivatives and polynomial multiples of $ f $ are in $ L^p(\mathbb{R}^d, \rho) $, and provide a single necessary-and-sufficient condition.
• To obtain dimension-independent conditions for the image, leveraging Nelson's hypercontractivity theorem and its implications for infinite-dimensional limits.

Proposed method

• Derives a family of inversion formulas for the Segal-Bargmann transform that can be tuned to optimize estimates for each $ p \in (1, \infty) $.
• Applies Hölder's inequality to the integral representation $ Sf(z) = (2\pi)^{-d/2} \int_{\mathbb{R}^d} e^{-(z-x)^2/2} f(x) \, dx $ to derive growth bounds on $ Sf(z) $ in terms of $ x $ and $ y $, where $ z = x + iy $.
• Uses the dual $ L^{p'} $-boundedness of the kernel $ e^{z \cdot x} $ to ensure convergence and holomorphy of $ Sf $.
• Establishes a necessary condition: $ |Sf(x+iy)| \leq C e^{y^2/2} e^{x^2/(2(p-1))} $, and a sufficient condition: $ \int_{\mathbb{C}^d} |F(x+iy)| e^{-y^2/2} e^{-x^2/(2(p-1))} \, dx\,dy < \infty $.
• Applies the ground state transformation to reframe the $ L^2 $-based Segal-Bargmann transform in terms of Gaussian measure, enabling extension to $ L^p $-spaces.
• Uses Nelson's hypercontractivity theorem to derive dimension-independent necessary and sufficient conditions for $ 1 < p \leq 2 $ and $ 2 \leq p < \infty $, respectively.

Experimental results

Research questions

• RQ1 What are the precise growth conditions on a holomorphic function that ensure it is the Segal-Bargmann transform of some $ f \in L^p(\mathbb{R}^d, \rho) $ for $ p \neq 2 $?
• RQ2 Can a single necessary-and-sufficient condition be given for the image of the $ L^p $-Schwartz space under the Segal-Bargmann transform?
• RQ3 Are there dimension-independent conditions characterizing the image of $ L^p(\mathbb{R}^d, \rho) $ under the Segal-Bargmann transform?
• RQ4 How do the growth rates in the real and imaginary directions of $ z \in \mathbb{C}^d $ depend on $ p $, and what is the sharpness of these bounds?
• RQ5 To what extent do hypercontractivity and logarithmic Sobolev inequalities inform the structure of the image in $ L^p $-spaces with Gaussian measure?

Key findings

• For $ f \in L^p(\mathbb{R}^d, \rho) $, the Segal-Bargmann transform satisfies the necessary growth bound $ |Sf(x+iy)| \leq C e^{y^2/2} e^{x^2/(2(p-1))} $, with the $ x $-dependence depending critically on $ p $.
• A holomorphic function $ F $ is in the image of $ L^p(\mathbb{R}^d, \rho) $ if $ \int_{\mathbb{C}^d} |F(x+iy)| e^{-y^2/2} e^{-x^2/(2(p-1))} \, dx\,dy < \infty $, which is only polynomially stronger than the necessary condition.
• For the $ L^p $-Schwartz space, a single necessary-and-sufficient condition is derived, generalizing Bargmann’s $ p=2 $ result to all $ p \in (1, \infty) $.
• The sufficient condition is sharp in the sense that if $ |F(x+iy)| \leq C e^{y^2/2} e^{x^2/(2(p-1))} (1+|x|)^{-d-\varepsilon}(1+|y|)^{-d-\varepsilon} $, then $ F $ is in the image of $ L^p(\mathbb{R}^d, \rho) $.
• Nelson’s hypercontractivity theorem yields dimension-independent necessary conditions for $ 1 < p \leq 2 $ and sufficient conditions for $ 2 \leq p < \infty $, valid even in the infinite-dimensional limit.
• The results are consistent with the ground state transformation, which reinterprets the Segal-Bargmann transform as a unitary map from $ L^2(\mathbb{R}^d, \rho) $ to $ \mathcal{H}L^2(\mathbb{C}^d, \mu) $, enabling extension to $ L^p $-spaces with Gaussian measure.