Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On multilinear determinant functionals


Author: Philip T. Gressman
Journal: Proc. Amer. Math. Soc. 139 (2011), 2473-2484
MSC (2010): Primary 28A75, 47G10; Secondary 42B10
DOI: https://doi.org/10.1090/S0002-9939-2010-10656-1
Published electronically: December 6, 2010
MathSciNet review: 2784813
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers the problem of $ L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin and by Bak, Oberlin, and Seeger.


References [Enhancements On Off] (What's this?)

  • [1] Jong-Guk Bak, Daniel M. Oberlin, and Andreas Seeger, Restriction of Fourier transforms to curves. II. Some classes with vanishing torsion, J. Aust. Math. Soc. 85 (2008), 1-28. MR 2460861 (2010b:42011)
  • [2] William Beckner, Geometric inequalities in Fourier anaylsis, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 1995, pp. 36-68. MR 1315541 (95m:42004)
  • [3] J. Bennett and N. Bez, Nonlinear geometric inequalities via induction-on-scales. Preprint. arXiv:0906.2064.
  • [4] J. Bennett, N. Bez, and A. Carbery, Heat-flow monotonicity reated to the Hausdorff-Young inequality, Bull. Lond. Math. Soc. 41 (2009), no. 6, 971-979. MR 2575327
  • [5] J. Bennett, N. Bez, A. Carbery, and D. Hundertmark, Heat-flow monotonicity of Strichartz norms, Anal. PDE 2 (2009), no. 2, 147-158. MR 2547132 (2010j:35418)
  • [6] Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287-299 (errata insert). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III. MR 0361607 (50:14052)
  • [7] Michael Christ, Estimates for the $ k$-plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891-910. MR 763948 (86k:44004)
  • [8] Michael Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), no. 1, 223-238. MR 766216 (87b:42018)
  • [9] Michael Christ, Convolution, curvature, and combinatorics: a case study, Internat. Math. Res. Notices (1998), no. 19, 1033-1048. MR 1654767 (2000a:42026)
  • [10] S. W. Drury and B. P. Marshall, Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 111-125. MR 764500 (87b:42019)
  • [11] S. W. Drury and B. P. Marshall, Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 541-553. MR 878901 (88e:42026)
  • [12] S. W. Drury, Estimates for a multilinear form on the sphere, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 3, 533-537. MR 0957258 (89m:42011)
  • [13] M. Burak Erdoğan and Richard Oberlin, Estimates for the $ X$-ray transform restricted to $ 2$-manifolds, Rev. Mat. Iberoam. 26 (2010), no. 1, 91-114. MR 2666309
  • [14] Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. MR 0257819 (41:2468)
  • [15] Philip Gressman, Sharp $ {L}^p - {L}^q$ estimates for generalized $ k$-plane transforms, Adv. Math. 214 (2007), no. 1, 344-365. MR 2348034 (2008m:47047)
  • [16] A. Iosevich and E. Sawyer, Maximal averages over surfaces, Adv. Math. 132 (1997), no. 1, 46-119. MR 1488239 (99b:42023)
  • [17] Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179-208. MR 1069246 (91i:42014)
  • [18] Alexander Nagel, Andreas Seeger, and Stephen Wainger, Averages over convex hypersurfaces, Amer. J. Math. 115 (1993), no. 4, 903-927. MR 1231151 (94m:42033)
  • [19] Daniel M. Oberlin, Convolution with measures on hypersurfaces, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 517-526. MR 1780502 (2001j:42014)
  • [20] D. H. Phong, E. M. Stein, and Jacob Sturm, Multilinear level set operators, oscillatory integral operators, and Newton polyhedra, Math. Ann. 319 (2001), no. 3, 573-596. MR 1819885 (2002f:42019)
  • [21] Elena Prestini, Restriction theorems for the Fourier transform to some manifolds in $ {\bf R}\sp{n}$ Harmonic analysis in Euclidean spaces, (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 101-109. MR 545244 (81d:42028)
  • [22] Terence Tao and James Wright, $ {L}\sp p$ improving bounds for averages along curves, J. Amer. Math. Soc. 16 (2003), no. 3, 605-638. MR 1969206 (2004j:42005)
  • [23] Stefán Ingi Valdimarsson, A multilinear generalisation of the Hilbert transform and fractional integration. Preprint.
  • [24] Antoni Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189-201. MR 0387950 (52:8788)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28A75, 47G10, 42B10

Retrieve articles in all journals with MSC (2010): 28A75, 47G10, 42B10


Additional Information

Philip T. Gressman
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19103
Email: gressman@math.upenn.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10656-1
Received by editor(s): April 5, 2010
Received by editor(s) in revised form: June 20, 2010
Published electronically: December 6, 2010
Additional Notes: The author was supported in part by NSF Grant DMS-0850791.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society