Arkiv för Matematik

  • Ark. Mat.
  • Volume 27, Number 1-2 (1989), 189-210.

Regularity of averages over hypersurfaces

Lennart Börjeson

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Averages over smooth measures on smooth compact hypersurfaces in Rn are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for example Lp estimates of multiparameter maximal functions over compact hypersurfaces.

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Ark. Mat., Volume 27, Number 1-2 (1989), 189-210.

Received: 1 August 1988
First available in Project Euclid: 31 January 2017

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1989 © Institut Mittag-Leffler


Börjeson, Lennart. Regularity of averages over hypersurfaces. Ark. Mat. 27 (1989), no. 1-2, 189--210. doi:10.1007/BF02386371.

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  • Benedek A. and Panzone R., The spaces Lp, with mixed norm, Duke Math. J. 28 (1961), 301–324.
  • Bergh, J. and Löfström, J., Interpolation spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
  • Börjeson, L., Mixed norm estimates for certain means, to appear in Trans. Amer. Math. Soc.
  • Bourgain, J., Estimations de certaines fonctions maximales, C.R. Acad. Sc. Paris 301 (1985), 499–502.
  • Bourgain, J., On the spherical maximal function in the plane, Preprint IHES, June 1985.
  • Bourgain, J., Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85.
  • Carlsson, H. and Sjögren, P., Estimates for maximal functions along hypersurfaces, Ark. Mat. 25 (1987), 1–14.
  • Carlsson, H., Sjögren, P. and Strömberg, J.-O., Multiparameter maximal functions along dilation-invariant hypersurfaces, Trans. Amer. Math. Soc. 292 (1985), 335–343.
  • Colzani, L., Regularity of spherical means and localization of spherical harmonic expansions, J. Austral. Math. Soc. Ser. A 41 (1986), 287–297.
  • Cowling, M. and Mauceri, G., Inequalities for some maximal functions. II, Trans. Amer. Math. Soc. 296 (1986), 341–365.
  • Gel’fand, I. M. and Shilov, G. E., Generalized functions, Vol. 1. Academic Press, New York-London, 1966.
  • Greenleaf, A., Principal curvature and harmonic analysis, Indiana Math. J. 30 (1981), 519–537.
  • Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Amer. Math. Soc. Coll., Providence, 1957.
  • Littman, M., Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766–770.
  • Oberlin, D. M. and Stein, E. M., Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), 641–650.
  • Peyrière J. and Sjölin, P., Regularity of spherical means, Ark. Mat. 16, (1978), 117–126.
  • Rubio De Francia, J. L., Maximal functions and Fourier transforms, Duke Math. J. 53 (1986), 395–404.
  • Sjölin, P., Lipschitz continuity of spherical means, in Linear spaces and approximation, Proc. Conf. Oberwolfach, 1977, Ed. by P. L. Butzer and B. Sz.-Nagy, Birkhäuser, 1978, 229–234.
  • Sjölin, P., Regularity and integrability of spherical means, Monatshefte f. Math. 96 (1983), 277–291.
  • Sjölin, P., Regularity and integrability properties of spherical means, Proc. of the Nineteenth Nordic Congress of Mathematicians, Reykjavík, 1984, 215–218.
  • Sjölin, P., Regularity properties of spherical means, in The Proceedings of Alfred Haar Memorial Conference, Budapest, 1985, Colloq. Math. Soc. János Bolyai 49, 823–846.
  • Sjölin, P., Norm inequalities for spherical means, Monatshefte f. Math. 100 (1985), 153–161.
  • Sogge, C. D., On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators, Trans. Amer. Math. Soc. 304 (1987), 733–749.
  • Sogge, C. D. and Stein, E. M., Averages of functions over hypersurfaces in Rn, Invent. Math. 82 (1985), 543–556.
  • Sogge, C. D. and Stein, E. M., Averages over hypersurfaces: II, Invent. Math. 86 (1986), 233–242.
  • Stein, E. M., Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.
  • Stein, E. M., Maximal functions: Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174–2175.
  • Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239–1295.
  • Stein, E. M. and Weiss G., Introduction to Fourier analysis in Euclidean spaces, Princeton University Press, Princeton, 1971.
  • Svensson, E., Fourier-Stieltjestransformer av mått på developpabla ytor, Mimeographed manuscript, Uppsala, May 1972.
  • Triebel, H., Interpolation theory, function, differential operators, North-Holland, Amsterdam, New York, Oxford, 1978.
  • Wainger, S., Averages and singular integrals over lower dimensional sets, in Beijing Lectures in Harmonic Analysis (ed. E. M. Stein), Annals Math. Studies, No. 112, Princeton Univ. Press, 1986, pp. 357–421.
  • Watson, G. N., Theory of Bessel functions, Cambridge University Press, Cambridge, 1944.