Arkiv för Matematik

  • Ark. Mat.
  • Volume 41, Number 2 (2003), 281-294.

Multipliers of spherical harmonics and energy of measures on the sphere

Kathryn E. Hare and Maria Roginskaya

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Abstract

We consider the operator, f(Δ) for Δ the Laplacian, on spaces of measures on the sphere in Rd, show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for the L2-norm of f(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure on Rd with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.

Note

This research was done while the first author enjoyed the hospitality of the Department of Mathematics of Göteborg University and Chalmers Institute of Technology. It was supported in part by NSERC and the Swedish natural sciences research council.

Article information

Source
Ark. Mat., Volume 41, Number 2 (2003), 281-294.

Dates
Received: 18 February 2002
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898806

Digital Object Identifier
doi:10.1007/BF02390816

Mathematical Reviews number (MathSciNet)
MR2011922

Zentralblatt MATH identifier
1034.43007

Rights
2003 © Institut Mittag-Leffler

Citation

Hare, Kathryn E.; Roginskaya, Maria. Multipliers of spherical harmonics and energy of measures on the sphere. Ark. Mat. 41 (2003), no. 2, 281--294. doi:10.1007/BF02390816. https://projecteuclid.org/euclid.afm/1485898806


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References

  • Alexsandrov, A. B., Function theory in the ball, in Current Problems in Mathematics, Fundamental Directions, Vol. 8 (Gamkrelidze, R. V., ed.), pp. 115–190, 274, Akad. Nauk SSSR, Vsesoyonz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985 (Russian). English transl.: in Several Complex Variables II (Khenkin, G. M. and Vitushkin, A. G., eds.), pp. 107–178, Springer-Verlag, New York, 1994.
  • Bourgain, J., Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), 193–201.
  • Davies, E. B., Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge Univ. Press, Cambridge, 1989.
  • Doubtsov, E., Singular measures with small H(p, q)-projections, Ark. Mat. 36, (1998), 355–361.
  • Falconer, K., Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 1990.
  • Hare, K. E. and Roginskaya, M., A Fourier series formula for energy of measures with applications to Riesz products, Proc. Amer. Math. Soc. 131 (2003), 165–174.
  • Mattila, P., Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34 (1987), 207–228.
  • Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Adv. Math. 44, Cambridge Univ. Press, Cambridge, 1995.
  • Mattila, P. and Sjölin, P., Regularity of distance measures and sets, Math. Nachr. 204 (1999), 157–162.
  • Rubin, B., Fractional Integrals and Potentials, Pitman Monographs 82, Addison-Wesley, Essex, 1996.
  • Rudin, W., Function Theory in the Unit Ball ofCn, Springer-Verlag, New York, 1980.
  • Sjölin, P. and Soria, F., Some remarks on restrictions of the Fourier transform for general measures, Publ. Math. 43 (1999), 655–664.
  • Stein, E. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971.
  • Wolff, T., Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547–567.