On adapted coordinate systems

Authors:
Isroil A. Ikromov and Detlef Müller

Journal:
Trans. Amer. Math. Soc. **363** (2011), 2821-2848

MSC (2000):
Primary 35D05, 35D10, 35G05

DOI:
https://doi.org/10.1090/S0002-9947-2011-04951-2

Published electronically:
January 26, 2011

MathSciNet review:
2775788

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Abstract | References | Similar Articles | Additional Information

Abstract: The notion of an adapted coordinate system for a given real-analytic function, introduced by V. I. Arnol'd, plays an important role, for instance, in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A. N. Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components. Varchenko's proof is based on a two-dimensional resolution of singularities result.

In this article, we present a more elementary approach to these results, which is based on the Puiseux series expansion of roots of the given function. This approach is inspired by the work of D. H. Phong and E. M. Stein on the Newton polyhedron and oscillatory integral operators. It applies to arbitrary real-analytic functions, and even to arbitrary smooth functions of finite type. In particular, we show that Varchenko's conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions, even in the smooth, finite type setting. For analytic functions, a construction of adapted coordinates by means of Puiseux series expansions of roots has already been carried out in work by D. H. Phong, E. M. Stein and J. A. Sturm on the growth and stability of real-analytic function, as we learned after the completion of this paper. In contrast to their work, however, our proof more closely follows Varchenko's algorithm for the construction of an adapted coordinate system, which turns out to be useful for the extension to the smooth setting.

**1.**V. I. Arnol′d,*Remarks on the method of stationary phase and on the Coxeter numbers*, Uspehi Mat. Nauk**28**(1973), no. 5(173), 17–44 (Russian). MR**0397777****2.**V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko,*Singularities of differentiable maps. Vol. II*, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR**966191****3.**Lars Hörmander,*The analysis of linear partial differential operators. I*, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR**1996773****4.**I. A. Ikromov, M. Kempe, and D. Müller,*Estimates for maximal functions associated with hypersurfaces in and related problems of harmonic analysis*, Acta Math.**204**(2010), 151-271.**5.**D. H. Phong and E. M. Stein,*The Newton polyhedron and oscillatory integral operators*, Acta Math.**179**(1997), no. 1, 105–152. MR**1484770**, https://doi.org/10.1007/BF02392721**6.**D. H. Phong, E. M. Stein, and J. A. Sturm,*On the growth and stability of real-analytic functions*, Amer. J. Math.**121**(1999), no. 3, 519–554. MR**1738409****7.**Elias M. Stein,*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192****8.**A. N. Varchenko,*Newton polyhedra and estimates of oscillating integrals*, Funct. Anal. and Appl.**10(3)**(1976), 175-196.

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Additional Information

**Isroil A. Ikromov**

Affiliation:
Department of Mathematics, Samarkand State University, University Boulevard 15, 703004, Samarkand, Uzbekistan

Email:
ikromov1@rambler.ru

**Detlef Müller**

Affiliation:
Mathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Straße 4, D-24098 Kiel, Germany

Email:
mueller@math.uni-kiel.de

DOI:
https://doi.org/10.1090/S0002-9947-2011-04951-2

Keywords:
Oscillatory integral,
Newton diagram

Received by editor(s):
April 8, 2008

Received by editor(s) in revised form:
September 23, 2008

Published electronically:
January 26, 2011

Additional Notes:
We acknowledge the support for this work by the Deutsche Forschungsgemeinschaft.

Article copyright:
© Copyright 2011
American Mathematical Society