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Transactions of the American Mathematical Society

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$ L^{p}$ boundedness of maximal averages over hypersurfaces in $ \mathbb{R}^{3}$

Author: Michael Greenblatt
Journal: Trans. Amer. Math. Soc. 365 (2013), 1875-1900
MSC (2010): Primary 42B20
Published electronically: September 19, 2012
MathSciNet review: 3009647
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Abstract: Extending the methods developed in the author's recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in two variables, an $ L^p$ boundedness theorem is proven for maximal operators over hypersurfaces in $ \mathbb{R}^3$ when $ p > 2.$ When the best possible $ p$ is greater than $ 2$, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Müller (2010).

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  • [B] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1986), 69-85. MR 874045 (88f:42036)
  • [CoMa] M. Cowling and G. Mauceri, Inequalities for some maximal functions. II, Trans. Amer. Math. Soc. 298 (1986), no. 1, 341-365. MR 837816 (87m:42013)
  • [G1] M. Greenblatt, Maximal averages over hypersurfaces and the Newton polyhedron, submitted.
  • [Gr] A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), no. 4, 519-537. MR 620265 (84i:42030)
  • [IkKeMu1] I. Ikromov, M. Kempe, and D. Müller, Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces (English summary) Duke Math. J. 126 (2005), no. 3, 471-490. MR 2120115 (2005k:42050)
  • [IkKeMu2] I. Ikromov, M. Kempe, and D. Müller, Estimates for maximal functions associated to hypersurfaces in $ R^{3}$ and related problems of harmonic analysis, Acta Math. 204 (2010), no. 2, 151-271. MR 2653054 (2011i:42026)
  • [IkMu] I. Ikromov and D. Müller, On adapted coordinate systems, Trans. Amer. Math. Soc., 363 (2011), 2821-2848. MR 2775788
  • [IoSa1] A. Iosevich and E. Sawyer, Oscillatory integrals and maximal averages over homogeneous surfaces, Duke Math. J. 82 no. 1 (1996), 103-141. MR 1387224 (97f:42035)
  • [IoSa2] A. Iosevich and E. Sawyer, Maximal averages over surfaces, Adv. Math. 132 (1997), no. 1, 46-119. MR 1488239 (99b:42023)
  • [NaSeWa] A. Nagel, A. Seeger, and S. Wainger, Averages over convex hypersurfaces, Amer. J. Math. 115 (1993), no. 4, 903-927. MR 1231151 (94m:42033)
  • [PSt] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), 107-152. MR 1484770 (98j:42009)
  • [So] C. Sogge, Maximal operators associated to hypersurfaces with one nonvanishing principal curvature (English summary) in Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), 317-323, Stud. Adv. Math., CRC, Boca Raton, FL, 1995. MR 1330250 (96e:42014)
  • [SoSt] C. Sogge and E. Stein, Averages of functions over hypersurfaces in $ R^{n}$, Invent. Math. 82 (1985), no. 3, 543-556. MR 811550 (87d:42030)
  • [St1] E. Stein, Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174-2175. MR 0420116 (54:8133a)
  • [St2] E. Stein, Harmonic analysis; real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematics Series Vol. 43, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • [V] A. N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funktsional. Anal. i Priložen 10 (1976), 13-38. English translation in Functional Anal. Appl. 18 (1976), no. 3, 175-196. MR 0422257 (54:10248)

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Additional Information

Michael Greenblatt
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices, 851 S. Morgan Street, Chicago, Illinois 60607-7045

Received by editor(s): April 27, 2011
Published electronically: September 19, 2012
Additional Notes: This research was supported in part by NSF grant DMS-0919713
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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