Lp improving bounds for averages along curves

Journal of the American Mathematical Society

Volumn 16, Issue 3, 2003, Pages 605-638

  Tao, Terence ^a   Wright, James ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Averaging operators; Double fibration; Lp improving properties; Radon transforms

Indexed keywords

EID: 0037791689     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-03-00420-X     Document Type: Article

References (53)

1. J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69-85. MR 88f:42036
2. J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 1 (1991), 147-187. MR 92g:42010
3. A. Carbery, M. Christ, J. Wright, Multidimensional van der Corput and Sublevel set estimates, J. Amer. Math. Soc. 12 (1999), 981-1015. MR 2000h:42010
4. M. Christ, Weak type (1,1) bounds for rough operators, Annals of Math. 128 (1988), 19-42. MR 89m:42013
5. M. Christ, Hubert transforms along curves I: Nilpotent groups, Annals of Math. 122 (1985), 575-596. MR 87f:42039a
6. M. Christ, Convolution, Curvature, and Combinatorics, A Case Study, Internat. Math. Research Notices 19 (1998), 1033-1048. MR 2000a:42026
7. M. Christ, M. B. Erdogan, Mixed-norm estimates for a restricted x-ray transform, preprint.
8. M. Christ, A. Nagel, E.M. Stein, S. Wainger, Singular Radon transforms: Analysis and geometry, Ann. of Math. (2) 150 (1999), 489-577. MR 2000J:42023
9. M. Christ and J. L. Rubio de Francia, Weak-type (1,1) bounds for rough operators II, Invent. Math. 93 (1988): 225-237. MR 90d:42021
10. n, Func. Ann. Appl. 2 (1968), 219-229. MR 38:6522
11. I.M. Gelfand, M.I. Graev, Z.J. Shapiro, Differential forms and integral geometry, Func. Ann. Appl. 3 (1969), 101-114. MR 39:6232
12. p-boundedness of singular Radon transforms in codimension 1, Duke Math. J. 108 (2001), 363-393. MR 2002d:42012
13. A. Greenleaf, A. Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35-56. MR 95h:58130
14. A. Greenleaf, A. Seeger, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), 1077-1119. MR 99g:58120
15. A. Greenleaf, A. Seeger, Oscillatory and Fourier integral operators with degenerate canonical relations, preprint.
16. A. Greenleaf, A. Seeger, S. Wainger, Estimates for generalized Radon transforms in three and four dimensions, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), 243 - 254, Contemp. Math., 251. MR 2001j:58047
17. 4, Proc. Amer. Math. Soc. 127 (1999), 3533-3545. MR 2001a:44002
18. A. Greenleaf, G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), 205-240. MR 91b:58251
19. A. Greenleaf, G. Uhlmann Composition of some singular Fourier integral operators and estimates for the X-ray transform, I, Ann. Inst. Fourier (Grenoble) 40 (1990), 443-466. MR 91k:58126
20. A. Greenleaf, G. Uhlmann Composition of some singular Fourier integral operators and estimates for the X-ray transform, II, Duke Math. J. 64 (1991), 413-419. MR 93b:58146
21. V. Guillemin, S. Sternberg, Geometric asymptotics, Amer. Math. Soc. Surveys 14, Providence, RI, 1977. MR 58:24404
22. 2,1 Ann. of Math. Studies 121, Princeton Univ. Press, Princeton, 1989. MR 91k:58140
23. S. Helgason, A duality in integral geometry on symmetric spaces, in Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto 1965 (1966), 37-56. MR 37:4765
24. L. Hormander, Fourier integral operators, Acta Math. 127 (1971), 79-183. MR 52:9299
25. q estimates for singular integral operators, Proc. Symp. Pure Math. 23, American Mathematical Society, Providence RI, 1973. MR 50:10909
26. L.H. Loomis, H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961-962. MR 11:166d
27. A. Nagel, E. Stein, S. Wainger, Balls and metrics defined by vector fields I. Basic properties, Acta. Math. 155 (1985), 103-147. MR 86k:46049
28. J. D. McMichael, Damping oscillatory integrals with polynomial phase, Math. Scand. 73 (1993), 215-228. MR 95f:42020
29. D. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), 56-60. MR 88f:42033
30. D. Oberlin, Multilinear proofs for two theorems on circular averages, Colloq. Math. 63 (1992), 187-190. MR 93m:42005
31. D. Oberlin, A convolution estimates for a measure on a curve in R4, Proc. Amer. Math. Soc. 125 (1997), 1355-1361. MR 97g:42009
32. D. Oberlin, A convolution estimates for a measure on a curve in R4 II, Proc. Amer. Math. Soc. 127 (1999), 217-221. MR 2000f:42007
33. D. Oberlin, An estimate for a restricted x-ray transform, Canad. Math. Bull. 43 (2000), 472-476. MR 2001k:44004
34. D. Oberlin, H. Smith, C. Sogge, Averages over curves with torsion, Math Res. Lett. 5 (1998), 535-539. MR 99j:42021
35. n, to appear, Revista Iberoamericana.
36. D. Phong, Regularity of Fourier integral operators, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 862-874, Birkhuser, Basel, 1995. MR 97e:58216
37. D. Phong, Singular integrals and Fourier integral operators, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 286-320, Princeton Math. Ser., 42, Princeton Univ. Press, Princeton, NJ, 1995. MR 95k:42026
38. D. Phong, E.M. Stein, Hubert transforms, singular integrals, and Radon transforms I, Acta Math. 157 (1986), 99-157. MR 88i:42028a
39. D. Phong, E.M. Stein, Radon transforms and torsion, IMRN 1091, 49-60. MR 93g:58144
40. D. Phong, E.M. Stein, On a stopping process for oscillatory integrals, J. Geom. Anal. 4 (1994), 105-120. MR 95k:42025
41. D. Phong, E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994), 703-722. MR 96c:35206
42. D. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), 105-152. MR 98j:42009
43. F. Ricci, E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals II: Singular kernels supported on submanifolds, J. Funct. Anal. 78:56-84. MR 89g:42030
44. W. Schlag, A generalization of Bourgain's circular maximal theorem, J. Amer. Math. Soc. 10 (1997), 103-122. MR 97c:42035
45. p improving properties of measures supported on curves on the Heisenberg group, Studia Math. 132 (1999), 179-201. MR 2000c:43010
46. A. Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), 685-745. MR 94h:35292
47. A. Seeger, Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (1998), 869-897. MR 99f:58202
48. E. M. Stein, Oscillatory integrals related to Radon-like transforms, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl. 1995, Special Issue, 535-551. MR 961:42013
49. T. H. Wolff, An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana. 11 (1995), 651-674. MR 96m:42034
50. T. H. Wolff, A Kakeya type problem for circles, Amer. J. Math. 119 (1997), 985-1026. MR 98m:42027
51. T. H. Wolff, A mixed norm estimate for the x-ray transform, Revista Mat. Iberoamericana. 14 (1998), 561-600. MR 2000j:44006
52. T. Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996), 129-162, Amer. Math. Soc., Providence, RI, 1999. MR 2000d:42010
53. p for large p, Geom. Funct. Anal. 10 (2000), 1237-1288. MR 2001k:42030