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

• Autorzy: Hans Triebel , Heike Winkelvoss
• 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.

Słowa kluczowe

EN

Hausdorff dimension; Hausdorff measure; function spaces

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 28A78: Hausdorff and packing measures
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 121
• Numer: 2
• Strony: 149-166

Daty

wydano

1996

otrzymano

1995-09-26

poprawiono

1996-06-10

Twórcy

autor

Hans Triebel

• Mathematisches Institut, Friedrich-Schiller-Universität, Jena D-07740, Jena, Germany,

autor

Heike Winkelvoss

• Mathematisches Institut, Friedrich-Schiller-Universität Jena, D-07740 Jena, Germany

Bibliografia

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