Uniform bounds for Fourier transforms of surface measures in R$^3$ with nonsmooth density
HTML articles powered by AMS MathViewer
- by Michael Greenblatt PDF
- Trans. Amer. Math. Soc. 368 (2016), 6601-6625 Request permission
Abstract:
We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in $\textbf {{R}}^3$. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of the $z$-axis and if $dS$ denotes the surface measure for this surface, then the measures under consideration are of the form $K(x,y)g(z) dS$ where $K(x,y)g(z)$ is supported near the origin and both $K(x,y)$ and $g(z)$ are allowed to have singularities. The estimates here generalize the previously known sharp uniform estimates for when $K(x,y)g(z)$ is smooth. The methods used in this paper involve an explicit two-dimensional resolution of singularities theorem, iterated twice, coupled with Van der Corput-type lemmas.References
- 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, DOI 10.1007/978-1-4612-3940-6
- Koji Cho, Joe Kamimoto, and Toshihiro Nose, Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude, J. Math. Soc. Japan 65 (2013), no. 2, 521–562. MR 3055595
- Michael Cowling and Giancarlo Mauceri, Oscillatory integrals and Fourier transforms of surface carried measures, Trans. Amer. Math. Soc. 304 (1987), no. 1, 53–68. MR 906805, DOI 10.1090/S0002-9947-1987-0906805-0
- J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974), 207–281. MR 405513, DOI 10.1002/cpa.3160270205
- M. Greenblatt, Estimates for Fourier transforms of surface measures in $\mathbf {R}^3$ with PDE applications, submitted.
- Michael Greenblatt, Resolution of singularities in two dimensions and the stability of integrals, Adv. Math. 226 (2011), no. 2, 1772–1802. MR 2737800, DOI 10.1016/j.aim.2010.09.003
- Michael Greenblatt, The asymptotic behavior of degenerate oscillatory integrals in two dimensions, J. Funct. Anal. 257 (2009), no. 6, 1759–1798. MR 2540991, DOI 10.1016/j.jfa.2009.06.015
- Isroil A. Ikromov, Michael Kempe, and Detlef Müller, Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces, Duke Math. J. 126 (2005), no. 3, 471–490. MR 2120115, DOI 10.1215/S0012-7094-04-12632-6
- Isroil A. Ikromov, Michael Kempe, and Detlef Müller, Estimates for maximal functions associated with hypersurfaces in $\Bbb R^3$ and related problems of harmonic analysis, Acta Math. 204 (2010), no. 2, 151–271. MR 2653054, DOI 10.1007/s11511-010-0047-6
- Isroil A. Ikromov and Detlef Müller, On adapted coordinate systems, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2821–2848. MR 2775788, DOI 10.1090/S0002-9947-2011-04951-2
- Isroil A. Ikromov and Detlef Müller, Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction, J. Fourier Anal. Appl. 17 (2011), no. 6, 1292–1332. MR 2854839, DOI 10.1007/s00041-011-9191-4
- Alex Iosevich and Eric Sawyer, Oscillatory integrals and maximal averages over homogeneous surfaces, Duke Math. J. 82 (1996), no. 1, 103–141. MR 1387224, DOI 10.1215/S0012-7094-96-08205-8
- A. Iosevich and E. Sawyer, Maximal averages over surfaces, Adv. Math. 132 (1997), no. 1, 46–119. MR 1488239, DOI 10.1006/aima.1997.1678
- J. Kamimoto and T. Nose, Toric resolution of singularities in a certain class of $C^{\infty }$ functions and asymptotic analysis of oscillatory integrals, preprint.
- V. N. Karpushkin, A theorem concerning uniform estimates of oscillatory integrals when the phase is a function of two variables, J. Soviet Math. 35 (1986), 2809-2826.
- V. N. Karpushkin, Uniform estimates of oscillatory integrals with parabolic or hyperbolic phases, J. Soviet Math. 33 (1986), 1159-1188.
- Ben Lichtin, Uniform bounds for two variable real oscillatory integrals and singularities of mappings, J. Reine Angew. Math. 611 (2007), 1–73. MR 2360603, DOI 10.1515/CRELLE.2007.073
- Walter Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766–770. MR 155146, DOI 10.1090/S0002-9904-1963-11025-3
- D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), no. 1, 105–152. MR 1484770, DOI 10.1007/BF02392721
- D. H. Phong and E. M. Stein, Damped oscillatory integral operators with analytic phases, Adv. Math. 134 (1998), no. 1, 146–177. MR 1612395, DOI 10.1006/aima.1997.1704
- Malabika Pramanik and Chan Woo Yang, Decay estimates for weighted oscillatory integrals in ${\Bbb R}^2$, Indiana Univ. Math. J. 53 (2004), no. 2, 613–645. MR 2060047, DOI 10.1512/iumj.2004.53.2388
- Christopher D. Sogge and Elias M. Stein, Averages of functions over hypersurfaces in $\textbf {R}^n$, Invent. Math. 82 (1985), no. 3, 543–556. MR 811550, DOI 10.1007/BF01388869
- 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
- A. N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Functional Anal. Appl. 18 (1976), no. 3, 175-196.
Additional Information
- Michael Greenblatt
- Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Office, 851 S. Morgan Street, Chicago, Illinois 60607-7045
- Email: greenbla@uic.edu
- Received by editor(s): January 30, 2014
- Received by editor(s) in revised form: May 14, 2014, and August 31, 2014
- Published electronically: November 12, 2015
- Additional Notes: This research was supported in part by NSF grant DMS-1001070
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6601-6625
- MSC (2010): Primary 42B20
- DOI: https://doi.org/10.1090/tran/6486
- MathSciNet review: 3461044