Abstract
Extending some resolution of singularities methods (Greenblatt in J Funct Anal 255(8):1957–1994, 2008) of the author, a generalization of a well-known theorem of Varchenko (Funct Anal Appl 18(3):175–196, 1976) relating decay of oscillatory integrals to the Newton polyhedron is proven. They are derived from analogous results for sublevel integrals, proven here. Varchenko’s theorem requires a certain nondegeneracy condition on the faces of the Newton polyhedron on the phase. In this paper, it is shown that the estimates of Varchenko’s theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold. Thus in problems where one wants to switch coordinates to a coordinate system where Varchenko’s estimates are valid, one has greater flexibility. Some additional estimates are also proven for more degenerate situations, including some too degenerate for the Newton polyhedron to give the optimal decay in the sense of Varchenko.
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Arnold V., Gusein-Zade S., Varchenko A.: Singularities of Differentiable Maps, vol. II. Birkhauser, Basel (1988)
Bierstone E., Milman P.: Resolution of singularities in Denjoy–Carleman classes. Selecta Math. (N.S.) 10(1), 1–28 (2004)
Christ M.: Hilbert transforms along curves. I. Nilpotent groups. Ann. Math. (2) 122(3), 575–596 (1985)
Christ M.: Convolution, curvature, and combinatorics, a case study. Int. Math. Res. Notices 19, 1033–1048 (1998)
Demailly J.-P., Kollar J.: Sem-continuity of complex singularity exponents and Kahler–Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4) 34(4), 525–556 (2001)
Greenleaf, A., Seeger, A.: Oscillatory and Fourier integral operators with degenerate canonical relation. Publicacions Matematiques special issue. In: Proceedings of the El Escorial Conference 2000, pp. 93–141 (2002)
Fedoryuk M.V.: The Saddle-Point Method. Nauka, Moscow (1977)
Ikromov, I., Müller, D.: On adapted coordinate systems. Trans. Am. Math. Soc. (to appear)
Karpushkin V.N.: Uniform estimates for volumes. Tr. Math. Inst. Steklova 221, 225–231 (1998)
Greenblatt M.: A coordinate-dependent local resolution of singularities and applications. J. Funct. Anal. 255(8), 1957–1994 (2008)
Greenblatt, M.: Resolution of singularities, asymptotic expansions of oscillatory integrals, and related Phenomena (submitted)
Greenblatt M.: Newton polygons and local integrability of negative powers of smooth functions in the plane. Trans. Am. Math. Soc. 358(2), 657–670 (2006)
Greenblatt M.: A direct resolution of singularities for functions of two variables with applications to analysis. J. Anal. Math. 92, 233–257 (2004)
Greenblatt M.: Sharp L 2 estimates for one-dimensional oscillatory integral operators with C ∞ phase. Am. J. Math. 127(3), 659–695 (2005)
Phong D.H., Stein E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179, 107–152 (1997)
Phong D.H., Stein E.M., Sturm J.: On the growth and stability of real-analytic functions. Am. J. Math. 121(3), 519–554 (1999)
Phong D.H., Sturm J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Ann. Math. (2) 152(1), 277–329 (2000)
Phong D.H., Sturm J.: On the algebraic constructibility of varieties of integrable rational functions on C n. Math. Ann. 323(3), 453–484 (2002)
Rychkov V.: Sharp L 2 bounds for oscillatory integral operators with C ∞ phases. Math. Zeitschrift 236, 461–489 (2001)
Seeger A.: Radon transforms and finite type conditions. J. Am. Math. Soc. 11(4), 869–897 (1998)
Stein, E.: Harmonic Analysis; Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematics Series, vol. 43, Princeton University Press, Princeton (1993)
Varchenko A.N.: Newton polyhedra and estimates of oscillatory integrals. Funct. Anal. Appl. 18(3), 175–196 (1976)
Vassiliev V.: The asymptotics of exponential integrals, Newton diagrams, and classification of minima. Func. Anal. Appl. 11, 163–172 (1977)
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This research was supported in part by NSF grant DMS-0654073.
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Greenblatt, M. Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Ann. 346, 857–895 (2010). https://doi.org/10.1007/s00208-009-0424-7
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DOI: https://doi.org/10.1007/s00208-009-0424-7