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

100%

Triebel H., Winkelvoss H.

Studia Mathematica

1996

tom 121

nr 2

149-166

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.

Spaces defined by the level function and their duals

100%

Sinnamon G.

Studia Mathematica

1994

tom 111

nr 1

19-52

The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of $L^p$. These domains are shown to be Banach spaces which, although closely tied to $L^p$ spaces, are not reflexive. A related construction is given which characterizes their dual spaces.

Ograniczanie wyników

2 Studia Mathematica

1 Sinnamon G.

1 Triebel H.

1 Winkelvoss H.

1 1996

1 1994

Wyniki wyszukiwania

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

Spaces defined by the level function and their duals