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
DOI:
https://doi.org/10.1090/S0002-9939-06-08356-0
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.
- J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147โ187. MR 1097257, DOI https://doi.org/10.1007/BF01896376
- M. Burak Erdogฬan, A note on the Fourier transform of fractal measures, Math. Res. Lett. 11 (2004), no. 2-3, 299โ313. MR 2067475, DOI https://doi.org/10.4310/MRL.2004.v11.n3.a3
- T. Mitsis, On a problem related to sphere and circle packing, J. London Math. Soc. (2) 60 (1999), no. 2, 501โ516. MR 1724841, DOI https://doi.org/10.1112/S0024610799007838
- 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. 95โ102. MR 717229, DOI https://doi.org/10.1007/BFb0061889
- Daniel M. Oberlin, An estimate for a restricted X-ray transform, Canad. Math. Bull. 43 (2000), no. 4, 472โ476. MR 1793949, DOI https://doi.org/10.4153/CMB-2000-055-8
- D. Oberlin Restricted Radon transforms and unions of hyperplanes Rev. Mat. Iberoamericana, to appear.
- 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
- Michel Talagrand, Sur la mesure de la projection dโun compact et certaines familles de cercles, Bull. Sci. Math. (2) 104 (1980), no. 3, 225โ231 (French, with English summary). MR 592470
- T. Wolff, Local smoothing type estimates on $L^p$ for large $p$, Geom. Funct. Anal. 10 (2000), no. 5, 1237โ1288. MR 1800068, DOI https://doi.org/10.1007/PL00001652
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
Email:
oberlin@math.fsu.edu
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.
