𝐿^{𝑝} improving bounds for averages along curves

Tao, Terence; Wright, James

doi:10.1090/S0894-0347-03-00420-X

$L^p$ improving bounds for averages along curves
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by Terence Tao and James Wright

J. Amer. Math. Soc. 16 (2003), 605-638

DOI: https://doi.org/10.1090/S0894-0347-03-00420-X

Published electronically: January 28, 2003

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Abstract:

We establish local $(L^p,L^q)$ mapping properties for averages on curves. The exponents are sharp except for endpoints.

References

J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 874045, DOI 10.1007/BF02792533
J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
Anthony Carbery, Michael Christ, and James Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), no. 4, 981–1015. MR 1683156, DOI 10.1090/S0894-0347-99-00309-4
Michael Christ, Weak type $(1,1)$ bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. MR 951506, DOI 10.2307/1971461
Michael Christ, Hilbert transforms along curves. I. Nilpotent groups, Ann. of Math. (2) 122 (1985), no. 3, 575–596. MR 819558, DOI 10.2307/1971330
Michael Christ, Convolution, curvature, and combinatorics: a case study, Internat. Math. Res. Notices 19 (1998), 1033–1048. MR 1654767, DOI 10.1155/S1073792898000610

Mixed-norm estimates for a restricted x-ray transform

Michael Christ, Alexander Nagel, Elias M. Stein, and Stephen Wainger, Singular and maximal Radon transforms: analysis and geometry, Ann. of Math. (2) 150 (1999), no. 2, 489–577. MR 1726701, DOI 10.2307/121088
Michael Christ and José Luis Rubio de Francia, Weak type $(1,1)$ bounds for rough operators. II, Invent. Math. 93 (1988), no. 1, 225–237. MR 943929, DOI 10.1007/BF01393693
I. M. Gel′fand and M. I. Graev, Line complexes in the space $C^{n}$, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 39–52 (Russian). MR 0238246
I. M. Gel′fand, M. I. Graev, and Z. Ja. Šapiro, Differential forms and integral geometry, Funkcional. Anal. i Priložen. 3 (1969), no. 2, 24–40 (Russian). MR 0244919
Michael Greenblatt, A method for proving $L^p$-boundedness of singular Radon transforms in codimension 1, Duke Math. J. 108 (2001), no. 2, 363–393. MR 1833395, DOI 10.1215/S0012-7094-01-10826-0
Allan Greenleaf and Andreas Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35–56. MR 1293873, DOI 10.1515/crll.1994.455.35
Allan Greenleaf and Andreas Seeger, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), no. 5, 1077–1119. MR 1646055, DOI 10.1353/ajm.1998.0036

Oscillatory and Fourier integral operators with degenerate canonical relations

Allan Greenleaf, Andreas Seeger, and Stephen Wainger, Estimates for generalized Radon transforms in three and four dimensions, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998) Contemp. Math., vol. 251, Amer. Math. Soc., Providence, RI, 2000, pp. 243–254. MR 1771272, DOI 10.1090/conm/251/03873
Allan Greenleaf, Andreas Seeger, and Stephen Wainger, On X-ray transforms for rigid line complexes and integrals over curves in $\textbf {R}^4$, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3533–3545. MR 1670367, DOI 10.1090/S0002-9939-99-05379-4
Allan Greenleaf and Gunther Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240. MR 1016420, DOI 10.1215/S0012-7094-89-05811-0
A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 443–466 (English, with French summary). MR 1070835, DOI 10.5802/aif.1220
Allan Greenleaf and Gunther Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. II, Duke Math. J. 64 (1991), no. 3, 415–444. MR 1141280, DOI 10.1215/S0012-7094-91-06422-7
Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965, DOI 10.1090/surv/014
Victor Guillemin, Cosmology in $(2 + 1)$-dimensions, cyclic models, and deformations of $M_{2,1}$, Annals of Mathematics Studies, vol. 121, Princeton University Press, Princeton, NJ, 1989. MR 999388, DOI 10.1515/9781400882410
Sigurđur Helgason, A duality in integral geometry on symmetric spaces, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 37–56. MR 0229191
Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
Walter Littman, $L^{p}-L^{q}$-estimates for singular integral operators arising from hyperbolic equations, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 479–481. MR 0358443
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
David McMichael, Damping oscillatory integrals with polynomial phase, Math. Scand. 73 (1993), no. 2, 215–228. MR 1269260, DOI 10.7146/math.scand.a-12467
Daniel M. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), no. 1, 56–60. MR 866429, DOI 10.1090/S0002-9939-1987-0866429-6
Daniel M. Oberlin, Multilinear proofs for two theorems on circular averages, Colloq. Math. 63 (1992), no. 2, 187–190. MR 1180631, DOI 10.4064/cm-63-2-187-190
Daniel M. Oberlin, A convolution estimate for a measure on a curve in $\mathbf R^4$, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1355–1361. MR 1363436, DOI 10.1090/S0002-9939-97-03716-7
Daniel M. Oberlin, A convolution estimate for a measure on a curve in $\mathbf R^4$. II, Proc. Amer. Math. Soc. 127 (1999), no. 1, 217–221. MR 1476381, DOI 10.1090/S0002-9939-99-04690-0
Daniel M. Oberlin, An estimate for a restricted X-ray transform, Canad. Math. Bull. 43 (2000), no. 4, 472–476. MR 1793949, DOI 10.4153/CMB-2000-055-8
Daniel Oberlin, Hart F. Smith, and Christopher D. Sogge, Averages over curves with torsion, Math. Res. Lett. 5 (1998), no. 4, 535–539. MR 1653332, DOI 10.4310/MRL.1998.v5.n4.a10

An x-ray estimate in $\mathbf {R}^n$

Duong H. Phong, Regularity of Fourier integral operators, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 862–874. MR 1403986
D. H. Phong, Singular integrals and Fourier integral operators, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 286–320. MR 1315553
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1986), no. 1-2, 99–157. MR 857680, DOI 10.1007/BF02392592
D. H. Phong and E. M. Stein, Radon transforms and torsion, Internat. Math. Res. Notices 4 (1991), 49–60. MR 1121165, DOI 10.1155/S1073792891000077
D. H. Phong and E. M. Stein, On a stopping process for oscillatory integrals, J. Geom. Anal. 4 (1994), no. 1, 105–120. MR 1274140, DOI 10.1007/BF02921595
D. H. Phong and E. M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994), no. 3, 703–722. MR 1307901, DOI 10.2307/2118622
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
Fulvio Ricci and Elias M. Stein, Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds, J. Funct. Anal. 78 (1988), no. 1, 56–84. MR 937632, DOI 10.1016/0022-1236(88)90132-2
W. Schlag, A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), no. 1, 103–122. MR 1388870, DOI 10.1090/S0894-0347-97-00217-8
Silvia Secco, $L^p$-improving properties of measures supported on curves on the Heisenberg group, Studia Math. 132 (1999), no. 2, 179–201. MR 1669682, DOI 10.4064/sm-132-2-179-201
Andreas Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), no. 3, 685–745. MR 1240601, DOI 10.1215/S0012-7094-93-07127-X
Andreas Seeger, Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (1998), no. 4, 869–897. MR 1623430, DOI 10.1090/S0894-0347-98-00280-X
Elias M. Stein, Oscillatory integrals related to Radon-like transforms, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), 1995, pp. 535–551. MR 1364908
Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. MR 1363209, DOI 10.4171/RMI/188
Thomas Wolff, A Kakeya-type problem for circles, Amer. J. Math. 119 (1997), no. 5, 985–1026. MR 1473067, DOI 10.1353/ajm.1997.0034
Thomas Wolff, A mixed norm estimate for the X-ray transform, Rev. Mat. Iberoamericana 14 (1998), no. 3, 561–600. MR 1681585, DOI 10.4171/RMI/245
Thomas Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996) Amer. Math. Soc., Providence, RI, 1999, pp. 129–162. MR 1660476
T. Wolff, Local smoothing type estimates on $L^p$ for large $p$, Geom. Funct. Anal. 10 (2000), no. 5, 1237–1288. MR 1800068, DOI 10.1007/PL00001652