Smooth solutions of quasianalytic or ultraholomorphic equations

Details Smooth solutions of quasianalytic or ultraholomorphic equations Smooth solutions of quasianalytic or ultraholomorphic equations

2008-09-11

arxiv.org/abs/0809.2057v1

Monatsh. Math. 160 (2010), 443-453

Mathematics

Classical Analysis and ODEs

8 pages

Scientific paper

10.1007/s00605-009-0108-0

In the first part of this work, we consider a polynomial $ \phi(x,y)=y^d+a_1(x)y^{d-1}+...+a_d(x) $ whose coefficients $ a_j $ belong to a Denjoy-Carleman quasianalytic local ring $ \mathcal{E}_1(M) $. Assuming that $ \mathcal{E}_1(M) $ is stable under derivation, we show that if $ h $ is a germ of $ C^\infty $ function such that $ \phi(x,h(x))=0 $, then $ h $ belongs to $ \mathcal{E}_1(M) $. This extends a well-known fact about real-analytic functions. We also show that the result fails in general for non-quasianalytic ultradifferentiable local rings. In the second part of the paper, we study a similar problem in the framework of ultraholomorphic functions on sectors of the Riemann surface of the logarithm. We obtain a result that includes suitable non-quasianalytic situations.

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