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

Hans Triebel; Heike Winkelvoss

Studia Mathematica (1996)

  • Volume: 121, Issue: 2, page 149-166
  • ISSN: 0039-3223

Abstract

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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 p , q s ( ℝ n ) (s ∈ ℝ, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.

How to cite

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Triebel, Hans, and Winkelvoss, Heike. "A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces." Studia Mathematica 121.2 (1996): 149-166. <http://eudml.org/doc/216348>.

@article{Triebel1996,
abstract = {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.},
author = {Triebel, Hans, Winkelvoss, Heike},
journal = {Studia Mathematica},
keywords = {Hausdorff dimension; Hausdorff measure; function spaces; Fourier analytical characterization; Lebesgue spaces; Besov spaces},
language = {eng},
number = {2},
pages = {149-166},
title = {A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces},
url = {http://eudml.org/doc/216348},
volume = {121},
year = {1996},
}

TY - JOUR
AU - Triebel, Hans
AU - Winkelvoss, Heike
TI - A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces
JO - Studia Mathematica
PY - 1996
VL - 121
IS - 2
SP - 149
EP - 166
AB - 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.
LA - eng
KW - Hausdorff dimension; Hausdorff measure; function spaces; Fourier analytical characterization; Lebesgue spaces; Besov spaces
UR - http://eudml.org/doc/216348
ER -

References

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  6. [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. Zbl0831.46028
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  8. [8] A. Jonsson and H. Wallin, Function Spaces on Subsets of ℝ n , Math. Reports 2, Part 1, Harwood Acad. Publ., London, 1984. Zbl0875.46003
  9. [9] A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math. 112 (1995), 285-300. Zbl0831.46029
  10. [10] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. 
  11. [11] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. 
  12. [12] H. Triebel and H. Winkelvoß, Intrinsic atomic characterizations of function spaces on domains, Math. Z. 221 (1996), 647-673. Zbl0843.46022
  13. [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. Zbl0851.46028
  14. [14] H. Winkelvoß, Function spaces related to fractals. Intrinsic atomic characterizations of function spaces on domains, Thesis, Jena, 1995. Zbl0880.46027

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