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)



Packing spheres and fractal Strichartz estimates in $ \mathbb{R}^d$ for $ d\geq 3$

Author: Daniel M. Oberlin
Journal: Proc. Amer. Math. Soc. 134 (2006), 3201-3209
MSC (2000): Primary 28A75, 35B45
Published electronically: May 11, 2006
MathSciNet review: 2231903
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove an estimate for the spherical average operator in $ \mathbb{R}^d$ if $ d\geq 3$. This leads to a lower bound for the Hausdorff dimension of unions of certain collections of spheres and to a Strichartz-type estimate for solutions of the wave equation.

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

  • 1. J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, 10.1007/BF01896376
  • 2. M. Burak Erdogan, A note on the Fourier transform of fractal measures, Math. Res. Lett. 11 (2004), no. 2-3, 299–313. MR 2067475, 10.4310/MRL.2004.v11.n3.a3
  • 3. T. Mitsis, On a problem related to sphere and circle packing, J. London Math. Soc. (2) 60 (1999), no. 2, 501–516. MR 1724841, 10.1112/S0024610799007838
  • 4. Daniel M. Oberlin, 𝐿^{𝑝}-𝐿^{𝑞} mapping properties of the Radon transform, Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981) Lecture Notes in Math., vol. 995, Springer, Berlin, 1983, pp. 95–102. MR 717229, 10.1007/BFb0061889
  • 5. Daniel M. Oberlin, An estimate for a restricted X-ray transform, Canad. Math. Bull. 43 (2000), no. 4, 472–476. MR 1793949, 10.4153/CMB-2000-055-8
  • 6. D. Oberlin Restricted Radon transforms and unions of hyperplanes Rev. Mat. Iberoamericana, to appear.
  • 7. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
  • 8. Michel Talagrand, Sur la mesure de la projection d’un compact et certaines familles de cercles, Bull. Sci. Math. (2) 104 (1980), no. 3, 225–231 (French, with English summary). MR 592470
  • 9. T. Wolff, Local smoothing type estimates on 𝐿^{𝑝} for large 𝑝, Geom. Funct. Anal. 10 (2000), no. 5, 1237–1288. MR 1800068, 10.1007/PL00001652

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A75, 35B45

Retrieve articles in all journals with MSC (2000): 28A75, 35B45

Additional Information

Daniel M. Oberlin
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510

Keywords: Spherical averages, Hausdorff dimension, Strichartz estimate
Received by editor(s): May 10, 2005
Published electronically: May 11, 2006
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.