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A microlocal approach to the enhanced Fourier-Sato transform in dimension one. (English) Zbl 1410.32007
Let $$V$$ be a one-dimensional complex vector space and $$V^*$$ its dual. The Fourier transform at the Weyl algebra level gives the isomorphism $$z\to-\partial_w$$, $$\partial_z\to w$$ which induces an equivalence between holonomic algebraic $${\mathcal D}$$-modules $${\mathcal M}$$ on $$V$$ and on $$V^*$$ denoted by $${\mathcal M}\to {}^L{\mathcal M}$$. At a microlocal level the Fourier transform is attached to the standard symplectic transform $$\chi:T^* V\to T^* V^*$$. This way a Legendre transform from Puiseux germs on $$V$$ to Puiseux germs on $$V^*$$ appears. The exponential factors of a holonomic algebraic $${\mathcal D}_V$$ module $${\mathcal M}$$ are Puiseux germs on $$V$$ which describe the growth of holomorphic solutions at irregular points. According to the stationary phase formula formulated in Theorem 1.1. the exponential factors of $$^L{\mathcal M}$$ are obtained via Legendre transform from the exponential factors of $${\mathcal M}$$.
The above-mentioned formula can be restated in terms of another realization of the Fourier transform, namely the Fourier-Sato transform for enhanced ind-sheaves. To do this the Riemann-Hilbert correspondence should be used. A microlocal proof of the stationary phase formula is given in the main result of the paper under consideration (Theorem 7.5.1.).

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 30E15 Asymptotic representations in the complex plane
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