estimates for overdetermined Radon transforms

Authors:
Luca Brandolini, Allan Greenleaf and Giancarlo Travaglini

Journal:
Trans. Amer. Math. Soc. **359** (2007), 2559-2575

MSC (2000):
Primary 42B10, 44A12

DOI:
https://doi.org/10.1090/S0002-9947-07-03953-0

Published electronically:
January 19, 2007

MathSciNet review:
2286045

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning bounds for convolution with all rotations of arc length measure on a fixed convex curve in . Estimates are obtained for averages over higher-dimensional convex (nonsmooth) hypersurfaces, smooth -dimensional surfaces, and nontranslation-invariant families of surfaces. We compare Ricci and Travaglini's approach, based on average decay of the Fourier transform, with an approach based on boundedness of Fourier integral operators, and show that essentially the same geometric condition arises in proofs using the two techniques.

**[BHI]**L. Brandolini, S. Hoffman and A. Iosevich,*Sharp rate of average decay of the Fourier transform of a bounded set*, Geom. Func. Anal.**13**(2003), 671-680. MR**2006553 (2004g:42015)****[BRT]**L. Brandolini, M. Rigoli and G. Travaglini,*Average decay of Fourier transforms and geometry of convex sets*, Rev. Mat. Iberoamericana**14**(1998), 519-560. MR**1681584 (2000a:42017)****[Bu]**M. Burak-Erdogan,*Mixed norm estimates for a restricted X-ray transform in and*, Internat. Math. Res. Notices (2001), 575-600. MR**1836731 (2002i:44003)****[BuC]**M. Burak-Erdogan and M. Christ,*Mixed norm estimates for a restricted X-ray transform*, J. Anal. Math.**87**(2002), 187-198. MR**1945281 (2003k:42030)****[C1]**M. Christ,*Estimates for the -plane transform*, Indiana Univ. Math. Jour.**33**(1984), 891-910. MR**0763948 (86k:44004)****[C2]**-,*Convolution, curvature and combinatorics: a case study*, Internat. Math. Res. Notices (19) (1998), 1033-1048. MR**1654767 (2000a:42026)****[CNSW]**M. Christ,A. Nagel, E. Stein and S. Wainger,*Singular and maximal Radon transforms: analysis and geometry.*, Ann. of Math.**150**(1999), 489-577. MR**1726701 (2000j:42023)****[D]**S. Dury,*Generalizations of Riesz potentials and estimates for certain -plane transforms*, Illinois Jour. Math.**28**(1984), 495-512. MR**0748958 (85h:44004)****[GS]**A. Greenleaf and A. Seeger,*Oscillatory and Fourier Integral operators with degenerate canonical relations*, Publicacions Matemátiques, Proc. of the 6th International Conf. on Harmonic Analysis and PDE (2002), 93-141. MR**1964817 (2004b:42027)****[GSW]**A. Greenleaf, A. Seeger and S. Wainger,*On X-ray transforms for rigid line complexes and integrals over curves in*, Proc. Amer. Math. Soc.**127**(1999), 3533-3545. MR**1670367 (2001a:44002)****[Gu]**V. Guillemin,*On some results of Gelfand in integral geometry*, Proc. Symp. Pure Math.**43**(1985), 149-155. MR**0812288 (87d:58137)****[GuSt]**V. Guillemin and S. Sternberg,*Geometric asymptotics*, AMS, Providence, 1977. MR**0516965 (58:24404)****[H1]**L. Hörmander,*Fourier integral operators I*, Acta Math.**127**(1971), 79-183. MR**0388463 (52:9299)****[H2]**-,*The analysis of linear partial differential operators, IV*, Springer-Verlag, New York, Berlin, 1985. MR**0781537 (87d:35002b)****[L]**W. Littman,*estimates for singular integral operators arising from hyperbolic equations*, Proc. Symp. Pure Math.**23**(1973), 479-481. MR**0358443 (50:10909)****[M]**B. Marshall,*Decay rates of Fourier transforms of curves*, Trans. Amer. Math. Soc.**310**(1988), 115-126. MR**0948194 (89m:42012)****[O]**D. Oberlin,*Convolution estimates for some measures on curves*, Proc. Amer. Math. Soc.**99**(1987), 56-60. MR**0866429 (88f:42033)****[P]**A.N. Podkorytov,*The asymptotic of a Fourier transform of a convex curve*, Vest. Leningr. Univ. Mat.**24**(1991), 57-65. MR**1166380 (93h:42019)****[PhS]**D. H. Phong and E.M. Stein,*Models of degenerate Fourier integral operators and Radon transforms*, Ann. Math.**140**(1994), 703-722. MR**1307901 (96c:35206)****[RT]**F. Ricci and G. Travaglini,*Convex curves, Radon transforms and convolution operators defined by singular measures*, Proc. A.M.S.**129**(2001), 1739-1744. MR**1814105 (2002i:42010)****[S]**A. Seeger,*Radon transforms and finite type conditions*, J. Amer. Math. Soc.**11**(1998), 869-897. MR**1623430 (99f:58202)****[Str]**R. Strichartz,*Convolutions with kernels having singularities on a sphere*, Trans. Amer. Math. Soc.**148**(1970), 461-471. MR**0256219 (41:876)****[TaW]**T. Tao and J. Wright,*improving bounds for averages along curves*, Jour. Amer. Math. Soc.**16**(2003), 605-638. MR**1969206 (2004j:42005)****[V]**A. Varchenko,*Number of lattice points in families of homothetic domains in*, Funk. An. App.**17**(1983), 1-6. MR**0705041 (85e:11077)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
42B10,
44A12

Retrieve articles in all journals with MSC (2000): 42B10, 44A12

Additional Information

**Luca Brandolini**

Affiliation:
Dipartimento di Ingegneria Gestionale e dell’Informazione, Università degli Studi di Bergamo, V.le G Marconi 5, 24044 Dalmine, Italy

Email:
brandolini@unibg.it

**Allan Greenleaf**

Affiliation:
Department of Mathematics, University of Rochester, Rochester, New York 14627

Email:
allan@math.rochester.edu

**Giancarlo Travaglini**

Affiliation:
Dipartimento di Statistica, Università di Milano-Bicocca, Edificio U7, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

Email:
giancarlo.travaglini@unimib.it

DOI:
https://doi.org/10.1090/S0002-9947-07-03953-0

Keywords:
Radon transform,
averages over curves,
$L^{p}$ improving

Received by editor(s):
December 16, 2003

Received by editor(s) in revised form:
February 7, 2005

Published electronically:
January 19, 2007

Additional Notes:
The second author was partially supported by a grant from the National Science Foundation.

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.