[Paper Review] Radon inversion formulas over local fields

Jonathan Wang|arXiv (Cornell University)|Mar 13, 2015

Mathematical Analysis and Transform Methods12 references3 citations

TL;DR

This paper establishes explicit inversion formulas for the Radon transform on local fields (real, complex, and non-Archimedean), proving the transform is an isomorphism between $K$-finite smooth function spaces with bounded and away-from-origin support. It provides explicit inverse formulas via Mellin transforms and Fourier analysis, resolving invertibility in the Archimedean case and relating to classical results like Černov's formula.

ABSTRACT

Let $F$ be a local field and $n\ge 2$ an integer. We study the Radon transform as an operator $M : \mathcal C_+ o \mathcal C_-$ from the space of smooth $K$-finite functions on $F^n \setminus \{0\}$ with bounded support to the space of smooth $K$-finite functions on $F^n \setminus \{0\}$ supported away from a neighborhood of $0$. These spaces naturally arise in the theory of automorphic forms. We prove that $M$ is an isomorphism and provide formulas for $M^{-1}$. In the real case, we show that when $K$-finiteness is dropped from the definitions, the analog of $M$ is not surjective.

Motivation & Objective

• To establish the invertibility of the Radon transform $M: \mathcal{C}_+ \to \mathcal{C}_-$ on local fields $F$ with $K$-finite smooth functions.
• To derive explicit formulas for the inverse operator $M^{-1}$ in both non-Archimedean and Archimedean (real and complex) cases.
• To clarify the relationship between the new inversion formulas and classical results, such as Černov's formula for Schwartz functions.
• To demonstrate that $M$ fails to be surjective when $K$-finiteness is removed, highlighting its essential role in invertibility.
• To provide a foundation for global intertwiners in automorphic forms by proving local invertibility of the Radon transform.

Proposed method

• Define the Radon transform $M$ as an operator between $K$-finite $C^\infty$ functions on $F^n \setminus \{0\}$ with bounded support ($\mathcal{C}_+$) and those supported away from a neighborhood of $0$ ($\mathcal{C}_-$).
• Use Fourier analysis and homogeneity to relate the Radon transform to the Mellin transform of radial distributions on $\mathbb{R}_{>0}$ in the real and complex cases.
• For non-Archimedean fields, relate the Radon transform to the Fourier transform on the additive group and derive an explicit inverse formula via $p$-adic integration.
• Prove that the inverse operator $M^{-1}$ is well-defined and continuous on each $K$-isotypic component of $\mathcal{C}_-$ using distributional convolution.
• Compute the Mellin transform of the kernel distribution in the real and complex cases using known integral formulas and Gamma function identities.
• Use analytic continuation to extend the inversion formulas beyond initial convergence regions, proving their validity for all $s \in \mathbb{C}$ away from poles.

Experimental results

Research questions

• RQ1 Is the Radon transform $M: \mathcal{C}_+ \to \mathcal{C}_-$ an isomorphism over local fields, including the real and complex cases?
• RQ2 Can an explicit inversion formula for $M^{-1}$ be derived in the non-Archimedean case, and how does it relate to Černov's formula for Schwartz functions?
• RQ3 What is the structure of the inverse operator $M^{-1}$ in the Archimedean case, and how does $K$-finiteness affect its invertibility?
• RQ4 What happens to the invertibility of $M$ if $K$-finiteness is dropped from the function space definitions?
• RQ5 How can the Mellin transform be used to characterize the inverse of the Radon transform in the real and complex settings?

Key findings

• The Radon transform $M: \mathcal{C}_+ \to \mathcal{C}_-$ is an isomorphism over all local fields $F$, including $\mathbb{R}$, $\mathbb{C}$, and non-Archimedean fields.
• In the non-Archimedean case, an explicit inversion formula for $M^{-1}$ is derived using Fourier transform techniques and shown to be equivalent to Černov's formula under $K$-finiteness.
• In the real case, $M^{-1}$ is given by convolution with a distribution on $\mathbb{R}_{>0}$ whose Mellin transform is explicitly computed as $\pi^{n-1}\frac{\Gamma(\frac{s+|p-q|}{2}-n+1)\Gamma(\frac{s-|p-q|}{2}-n+1)}{\Gamma(\frac{s+p+q}{2})\Gamma(\frac{s-p-q}{2}-n+1)}$.
• In the complex case, the inverse is similarly characterized via the Mellin transform of a radial distribution, with the same functional form as in the real case.
• The $K$-finiteness assumption is essential: when dropped, the analog of $M$ is not surjective, as shown in Corollary 4.7.4.
• The results provide a local foundation for the invertibility of global intertwiners in the theory of Eisenstein series and automorphic forms, as used in [DW].