Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees
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Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees Pekka Koskela1 · Zhuang Wang1 Received: 25 March 2019 / Accepted: 9 October 2019 / © The Author(s) 2019
Abstract In this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces. Keywords Besov-type space · Regular tree · Trace space · Dyadic norm Mathematics Subject Classification (2010) 46E35 · 30L99
1 Introduction Over the past two decades, analysis on general metric measure spaces has attracted a lot of attention, e.g., [2, 4, 12, 13, 15–17]. Especially, the case of a regular tree and its Cantor-type boundary has been studied in [3]. Furthermore, Sobolev spaces, Besov spaces and TriebelLizorkin spaces on metric measure spaces have been studied in [5, 25, 26] via hyperbolic fillings. A related approach was used in [23], where the trace results of Sobolev spaces and of related fractional smoothness function spaces were recovered by using a dyadic norm and the Whitney extension operator. Dyadic energy has also been used to study the regularity and modulus of continuity of space-filling curves. One of the motivations for this paper is the approach in [20]. Given a continuous g : S 1 → Rn , consider the dyadic energy
E (g; p, λ) :=
+∞ i=1
i
i
λ
2 j =1
|gIi,j − gIi,j |p .
Zhuang Wang
[email protected] Pekka Koskela [email protected] 1
Department of Mathematics and Statistics, University of Jyv¨askyl¨a, PO Box 35, FI-40014 Jyv¨askyl¨a, Finland
(1.1)
P. Koskela, Z. Wang
Here, {Ii,j : i ∈ N, j = 1, · · · , 2i } is a dyadic decomposition of S 1 such that for every fixed i ∈ N, {Ii,j : j = 1, · · · , 2i } is a family of arcs of length 2π/2i with j Ii,j = S 1 . The next generation is constructed in such a way that for each j ∈ {1, · · · , 2i+1 }, there exists a unique number k ∈ {1, · · · , 2i }, satisfying Ii+1,j ⊂ Ii,k . We denote this parent of Ii+1,j by Ii+1,j and set I1,j = S 1 for j = 1, 2. By gA , A ⊂ S 1 , we denote the mean value gA = −A g d H1 = H11(A) A g d H1 . One could expect to be able to use the energy Eq. 1.1 to characterize the trace spaces of some Sobolev spaces (with suitable weights) on the unit disk. On the contrary, the results in [23] suggest that the trace spaces of Sobolev spaces (with suitable weights) on the unit disk should be characterized by the energy E(g; p, λ) :=
+∞
i
iλ
2
|gIi,j − gIi,j −1 |p ,
(1.2)
j =1
i=1
where Ii,0 = Ii,2i , and the example g(x) = χI1,1 shows that E (g; p, λ) is not comparable to E(g; p, λ). Notice that the energies (1.1) and (1.2) can be viewed as dyadic energies on the boundary of a binary tree (2-regular tree). More precisely, for a 2-regular tree X in Section 2.1 with = log 2 in the metric (2.1), the measure ν on the boundary ∂X is the Hausdorff 1-measure by Proposition 2.10. Furthermore, there is a one-to-one map h from the dyadic decomposition of S 1 to the dyadic decompos
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