Abstract
In connection with fractal elliptic operators which will be considered in Chapter III we are particularly interested in d-sets and their perturbations, called (d, Ψ)-sets. A compact set Γ in \( {\mathbb{R}^n} \) is called a d-set (in our notation) or a (d, Ψ)-set if there is a Radon measure μ in \( {\mathbb{R}^n} \) with Γ =supp μ such that
respectively, whereB(γr) is a ball centred atγ ∈ Γ and of radius r. Naturally 0 ≤d ≤n and Ψ(r) is a perturbation where one might think typically of Ψ(r) = |log r| b for some b ∈ ℝ. However, in connection with fractal drums in the plane, it is reasonable and desirable to admit more general sets P and related measures µ with supp µ=Γ. This leads naturally to the question under what circumstances functions f belonging to, say, some Sobolev spaces H s p (ℝn) with s < 0 and 1 > p > ∞ have traces trΓon Ψ such that
for some r with, say,1≤r≤ ∞,and c >0. Problems of this type have been studied with great intensity for more than thirty years, pioneered by Maz’ya, and then by D. R. Adams, Hedberg, Netrusov, Verbitsky and many other mathematicians, usually in the context of (linear and non-linear) potential theory. This means, instead of (9.2), one asks for inequalities of type
where again Γ is the support of a given measure µ in \( {\mathbb{R}^n} \) and I s is the Riesz potential,
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© 2001 Springer Basel AG
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Triebel, H. (2001). Traces on sets, related function spaces and their decompositions. In: The Structure of Functions. Monographs in Mathematics, vol 97. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8257-6_9
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DOI: https://doi.org/10.1007/978-3-0348-8257-6_9
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9494-4
Online ISBN: 978-3-0348-8257-6
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