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Some phases of oscillatory integrals

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Abstract

The first example of a phase is presented for which Arhold’s conjecture on the validity of uniform estimates for oscillatory integrals with maximal singularity index is true, while his conjecture on the semicontinuity of the singularity index is false. A rough upper bound for the Milnor number such that the latter conjecture fails is obtained. The corresponding counterexample is simpler than Varchenko’s well-known counterexample to Arnold’s conjecture on the semicontinuity of the singularity index. This gives hope to decrease codimension and the Milnor number for which the conjecture on the semicontinuity of the singularity index fails.

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Authors and Affiliations

  1. Institute for Information Transmission Problems, Moscow, Russia

    V. N. Karpushkin

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  1. V. N. Karpushkin
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Correspondence to V. N. Karpushkin.

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__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 45, No. 2, pp. 91–93, 2011

Original Russian Text Copyright © by V. N. Karpushkin

This work was supported by grant NSh-8462.2010.1.

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Karpushkin, V.N. Some phases of oscillatory integrals. Funct Anal Its Appl 45, 154–156 (2011). https://doi.org/10.1007/s10688-011-0017-6

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  • DOI: https://doi.org/10.1007/s10688-011-0017-6

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