Karpushkin, V. N. Some phases of oscillatory integrals. (English. Russian original) Zbl 1271.42022 Funct. Anal. Appl. 45, No. 2, 154-156 (2011); translation from Funkts. Anal. Prilozh. 45, No. 2, 91-93 (2011). Summary: 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. MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) Keywords:oscillatory integral; phase; amplitude; volume PDF BibTeX XML Cite \textit{V. N. Karpushkin}, Funct. Anal. Appl. 45, No. 2, 154--156 (2011; Zbl 1271.42022); translation from Funkts. Anal. Prilozh. 45, No. 2, 91--93 (2011) Full Text: DOI References: [1] V. I. Arnold, Singularities of Caustics and Wave Fronts, Kluwer Acad. Publ., Dordrecht, 1990. [2] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps, vol. 1, Birkhäuser, Boston, MA, 1985. [3] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable map, vol. 2, Birkhäuser, Boston, MA, 1988. · Zbl 1297.32001 [4] A. N. Varchenko, Funkts. Anal. Prilozhen., 10:3 (1976), 13–38; English transl.: Functional Anal. Appl., 10:3 (1976), 175–196. [5] V. A. Vasil’ev, Funkts. Anal. Prilozhen., 11:3 (1977), 1–11; English transl.: Functional Anal. Appl., 11:3 (1977), 163–172. · Zbl 0368.34015 · doi:10.1007/BF01135526 [6] V. N. Karpushkin, Trudy Sem. Petrovsk., 9 (1983), 3–39. [7] V. N. Karpushkin, Trudy Sem. Petrovsk., 10 (1984), 150–169. [8] V. N. Karpushkin, Funkts. Anal. Prilozhen., 26:1 (1992), 59–61; English transl.: Functional Anal. Appl., 26:1 (1992), 46–48. · Zbl 0760.43004 · doi:10.1007/BF01077080 [9] V. N. Karpushkin, Trudy Mat. Inst. Steklova, 221 (1998), 225–231; English transl.: Proc. Steklov Inst. Math., 221 (1998), 214–220. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.