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1996 | 121 | 2 | 149-166
Tytuł artykułu

A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let Γ be a closed set in $ℝ^n$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_{1} > 0$ and $c_{2} > 0$ such that $c_{1}r^{d} ≤ µ (B(x,r)) ≤ c_{2}r^{d}$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_{p}(Γ)$, 0 < p ≤ ∞, with respect to that measure µ on the hand and the Fourier analytically defined Besov spaces $B^s_{p,q}(ℝ^n)$ (s ∈ ℝ, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.
Czasopismo
Rocznik
Tom
121
Numer
2
Strony
149-166
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-09-26
poprawiono
1996-06-10
Twórcy
autor
  • Mathematisches Institut, Friedrich-Schiller-Universität Jena, D-07740 Jena, Germany
Bibliografia
  • [1] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, 1985.
  • [2] K. J. Falconer, Fractal Geometry, Wiley, 1990.
  • [3] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.
  • [4] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
  • [5] A. B. Gulisashvili, Traces of differentiable functions on subsets of the euclidean space, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (1986), 52-66 (in Russian).
  • [6] A. Jonsson, Atomic decomposition of Besov spaces on closed sets, in: Function Spaces, Differential Operators and Non-Linear Analysis, Teubner-Texte Math. 133, Teubner, Leipzig, 1993, 285-289.
  • [7] A. Jonsson, Besov spaces on closed sets by means of atomic decompositions, preprint, Umeå, 1993.
  • [8] A. Jonsson and H. Wallin, Function Spaces on Subsets of $ℝ^n$, Math. Reports 2, Part 1, Harwood Acad. Publ., London, 1984.
  • [9] A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math. 112 (1995), 285-300.
  • [10] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
  • [11] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
  • [12] H. Triebel and H. Winkelvoß, Intrinsic atomic characterizations of function spaces on domains, Math. Z. 221 (1996), 647-673.
  • [13] H. Triebel and H. Winkelvoß, The dimension of a closed subset of $ℝ^n$ and related function spaces, Acta Math. Hungar. 68 (1995), 117-133.
  • [14] H. Winkelvoß, Function spaces related to fractals. Intrinsic atomic characterizations of function spaces on domains, Thesis, Jena, 1995.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv121i2p149bwm
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