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Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets

  • Pertti Mattila (a1)
Abstract

Let μ, be a positive Radon measure with compact support in the euclidean n-space ℝn. Introducing the Fourier transform

and the averages over the spheres

we can write the α-energy, 0 < α < n, of μ as

where the positive constants c1 and c2 depend only on n and α. The second equality is based on the Plancherel formula and the fact that where .

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References
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C.Carleson L.. Selected Problems on Exceptional Sets (Van Nostrand, 1967).
F1.Falconer K. J.. Geometry of Fractal Sets (Cambridge University Press, 1985).
F2.Falconer K. J.. On the Hausdorff dimension of distance sets. Mathematika, 32 (1985), 206212.
K1.Kahane J.-P.. Some Random Series of Functions (Heath Mathematical Monographs, 1968).
K2.Kahane J.-P.. Sur la dimensions des intersections, Aspects of Mathematics and its Applications, Barroso J. A. editor (Elsevier Science Publishers B.V., 1986, 419430).
KS.Kahane J.-P. and Salem R.. Ensembles parfaits et sèries trigonométriques (Hermann, 1963).
KR.Kaufmann R.. On the theorem of Jarnik and Besicovitch. Acta Arithmetica, 33 (1981). 265267.
L.Landkof N. S.. Foundations of Modern Potential Theory (Springer, 1972).
M1.Mattila P.. Hausdorff dimension and capacities of intersections of sets in n-space. Acta Math., 152 (1984), 77105.
M2.Mattila P., On the Hausdorff dimension and capacities of intersections. Mathematika, 32 (1985), 213217.
M.Muckenhoupt B.. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165 (1972), 207226.
SW.Stein E. M. and Weiss G.. Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, 1971).
W.Watson G. N.. Theory of Bessel Functions (Cambridge University Press, 1944).
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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