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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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