A microlocal approach to the enhanced Fourier-Sato transform in dimension one

TL;DR

This paper provides a microlocal proof of the stationary phase formula for the Fourier transform of holonomic D-modules on the affine line, linking exponential factors via Legendre transform using enhanced ind-sheaves.

Contribution

It introduces a microlocal approach to the Fourier-Sato transform, connecting exponential factors and Legendre transform through enhanced ind-sheaves and Riemann-Hilbert correspondence.

Findings

Microlocal proof of stationary phase formula

Connection between exponential factors and Legendre transform

Application of enhanced ind-sheaves in D-module analysis

Abstract

Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to $\mathcal{M}$ at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of $\mathcal{M}$ are obtained by Legendre transform from the exponential factors of $\mathcal{M}$. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.