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Resurgent analysis in several variables. I: General theory. (English) Zbl 0831.35021
Lumer, Günter (ed.) et al., Partial differential equations. Models in physics and biology. Contributions to the conference, held in Han-sur- Lesse, Belgium, in December 1993. Berlin: Akademie Verlag. Math. Res. 82, 351-377 (1994).
Resurgent analysis is a function theory of special character in which the functions depend on large (small) parameters called resurgent variables, appearing in the study of the asymptotic behavior of the solutions of PDE and Mathematical Physics. In the case of one resurgent variable the theory was developed by J. Ecalle during 1981-85. An excellent exposition of Ecalle’s theory is given by B. Malgrange [Enseign. Math., II. Sér. 31, 261-282 (1985; Zbl 0601.58043)], where it is shown that the resurgent function theory of one variable $$x$$ is essentially a generalization of the Borel-Laplace integral transformation of microfunctions of a dual variable $$s$$ [see also: J.-C. Toujeron, Lectures on Borel-Laplace transformations and resurgent functions, Preprint (1992)]. In the paper under review, the authors develop the case of several variables by sophisticated methods using sheaves integral transformations and a kind of microfunctions introduced here. In fact, this is an attempt of constructing resurgent function theory of several variables with the help of integral transformations denoted $$l_\varphi$$ acting on germs of ramifing analytic functions considered just as microfunctions. In the simplest case a resurgent function $$f(x)$$, $$x \in \mathbb{C}^n$$, is represented as follows: $$f(x) = \ell_\varphi (F(s,x))$$, where $$F(s,x)$$ is a homogeneous order $$- 1$$ exponential microfunction of the mentioned kind. It is proved that the action of $$\ell_\varphi$$ on the convolution $$F*G$$ is like the classical one and that $$\ell_\varphi$$ commutes with the operator $$\partial/ \partial s$$ in some sence. The main feature of the developed theory is that the study of the behaviour of the resurgent function $$f(x)$$ at infinity is reduced to the investigation of the corresponding microfunction $$F(s,x)$$ near its singularities (focal points).
For the entire collection see [Zbl 0809.00019].

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35C15 Integral representations of solutions to PDEs 35Q40 PDEs in connection with quantum mechanics 47A40 Scattering theory of linear operators 47F05 General theory of partial differential operators