# 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

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topTriebel, 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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- [10] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
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- [12] H. Triebel and H. Winkelvoß, Intrinsic atomic characterizations of function spaces on domains, Math. Z. 221 (1996), 647-673. Zbl0843.46022
- [13] H. Triebel and H. Winkelvoß, The dimension of a closed subset of ${\mathbb{R}}^{n}$ and related function spaces, Acta Math. Hungar. 68 (1995), 117-133. Zbl0851.46028
- [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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