[Paper Review] Fourier transforms of spherical distributions on compact symmetric spaces

Gestur Ólafsson, Henrik Schlichtkrull|ArXiv.org|Oct 1, 2008

Advanced Algebra and Geometry26 references3 citations

TL;DR

This paper extends Paley-Wiener theorems to $K$-invariant distributions on compact symmetric spaces $M = U/K$, proving that the Fourier transform of a distribution with small compact support extends to a holomorphic function of exponential type on $\mathfrak{a}_{\mathbb{C}}^*$, with the growth rate matching the support radius. The image of the Fourier transform is characterized as a Paley-Wiener space, and the singular support is linked to decay conditions on the transform, generalizing results from smooth functions to distributions.

ABSTRACT

In our previous articles "A local Paley-Wiener theorem for compact symmetric spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact symmetric spaces" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. We characterize the singular support of a distribution in terms of its Fourier transform. Finally, we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator.

Motivation & Objective

• To generalize the Paley-Wiener theorem from smooth $K$-invariant functions to $K$-invariant distributions on compact symmetric spaces of the form $M = U/K$.
• To characterize the image of the Fourier transform of such distributions as a space of holomorphic functions of exponential type.
• To relate the singular support of a distribution to decay properties of its Fourier transform, extending results from the smooth case.
• To establish a characterization of distributions of small support that lie in the range of a given invariant differential operator.

Proposed method

• Use the spherical Fourier transform on $\mathfrak{a}_{\mathbb{C}}^*$, where $\mathfrak{a}$ is a maximal abelian subspace of the non-compact part of the symmetric space.
• Leverage results from prior work on $K$-invariant smooth functions (from [27]) to extend the Paley-Wiener theorem to distributions via duality and analytic continuation.
• Define the Paley-Wiener space $\mathrm{PW}_r^*(\mathfrak{a})$ as the space of holomorphic functions on $\mathfrak{a}_{\mathbb{C}}^*$ satisfying exponential-type growth estimates with exponent $r$ matching the support radius.
• Apply Hörmander-type estimates to characterize singular support via decay conditions on the Fourier transform, specifically requiring bounds of the form $|F(\lambda)| \leq C_m(1+|\lambda|)^N e^{r|\mathrm{Im}(\lambda)|}$ under certain growth constraints on $|\mathrm{Im}(\lambda)|$.
• Use covering space techniques to extend results from semisimple groups to general compact symmetric spaces by lifting functions and distributions to a covering space $Z_0 \times U'/K'$, preserving $K$-invariance and Fourier transform structure.
• Establish a commutative diagram relating Fourier transforms on the original space and its cover, ensuring that the Paley-Wiener space is preserved and the transform is injective on $K$-invariant distributions.

Experimental results

Research questions

• RQ1 How does the Fourier transform of a $K$-invariant distribution on a compact symmetric space $M = U/K$ extend to a holomorphic function, and what growth conditions characterize its image?
• RQ2 What is the precise relationship between the support radius of a $K$-invariant distribution and the exponential type of its Fourier transform?
• RQ3 How can the singular support of a $K$-invariant distribution be characterized in terms of the decay and growth of its Fourier transform?
• RQ4 Can the Paley-Wiener theorem for smooth $K$-invariant functions be extended to the space of $K$-invariant distributions with compact support?
• RQ5 How can the range of an invariant differential operator on $K$-invariant distributions be characterized using the Fourier transform?

Key findings

• The Fourier transform of a $K$-invariant distribution with support in a ball of radius $r$ extends to a holomorphic function on $\mathfrak{a}_{\mathbb{C}}^*$ of exponential type with exponent $r$, matching the support radius.
• The image of the Fourier transform on $K$-invariant distributions with support in a ball of radius $r$ is precisely the Paley-Wiener space $\mathrm{PW}_r^*(\mathfrak{a})$, consisting of holomorphic functions satisfying $|F(\lambda)| \leq C_N(1+|\lambda|)^{-N}e^{r|\mathrm{Im}(\lambda)|}$ for all $\lambda \in \mathfrak{a}_{\mathbb{C}}^*$ and all $N \in \mathbb{Z}^+$.
• The singular support of a $K$-invariant distribution is contained in a closed ball of radius $r$ if and only if its Fourier transform satisfies the growth condition $|F(\lambda)| \leq C_m(1+|\lambda|)^N e^{r|\mathrm{Im}(\lambda)|}$ for all $\lambda$ with $|\mathrm{Im}(\lambda)| \leq m \log(1+|\lambda|)$, for some $N$ and all $m \in \mathbb{Z}^+$.
• The distributional Paley-Wiener theorem characterizes $K$-invariant distributions of small support as those whose Fourier transforms lie in $\mathrm{PW}_r^*(\mathfrak{a})$, with the exponent $r$ equal to the radius of the smallest closed ball containing the support.
• The results extend to all compact symmetric spaces via a covering space construction: $U/K$ is covered by $Z_0 \times U'/K'$, and the Paley-Wiener theorem holds for $U/K$ if it holds for the cover, preserving $K$-invariance and transform structure.
• The Fourier transform on $\mathrm{PW}_r^*(\mathfrak{a})$ is injective on $K$-invariant distributions, and functions in this space are uniquely determined by their restriction to the set of highest weights $\Lambda^+(U/K)$.