A trace theorem for Dirichlet forms on fractals^{∗}

Masanori Hino (Kyoto Univ.)

In this talk, I will report a trace theorem for self-similar Dirichlet forms on fractal sets to fractal subsets. A classical trace theorem for Besov spaces on the Euclidean space is the following.

Theorem. Let s > 0, 1 ≤ p < ∞, 1 ≤ q ≤ ∞, 0 ≤ d < n, β := s − (n − d)/p > 0. Then, the restriction map from S(ℝ^{n}) to S(ℝ^{d}) extends continuously to the trace operator tr: B_{p,q}^{s}(ℝ^{n}) → B_{p,q}^{β}(ℝ^{d}). Further, tr is surjective and has a bounded linear right inverse.

Trace theorems on ℝ^{n} to Ahlfors regular sets are also discussed in [4,6]. The differential structure of ℝ^{n} is crucial there. On the other hand, Jonsson [3] proved the following trace theorem.

Theorem. Let K be a 2-dim. Sierpinski gasket and I the base of the triangle. Let μ be the normalized Hausdorff measure on K and (E, F) the canonical Dirichlet form on L^{2}(K, μ). Then, there exists a trace operator from F to Λ_{2,2}^{β}(I), where β = d_{w}/2 − (d_{f} − d)/2 with d_{w} = log 5/ log 2, d_{f} = log 3/ log 2, and d = 1.

Here, for an Ahlfors regular metric measure space (X, d, μ), the Besov space Λ_{p,q}^{β}(X) is defined as follows. For 1 ≤ p < ∞, 1 ≤ q ≤ ∞, β ≥ 0 and m ∈ ℤ_{+}, set

a_{m}(β, f) := α^{mβ} (α^{md_{f}} ∫ ∫_{{d(x,y)<c_{0}α^{−m}}} |f(x) − f(y)|^{p}dμ(x)dμ(y))^{1/p} for f ∈ L^{p}(X, μ),

where 1 < α < ∞, 0 < c_{0} < ∞, and d_{f} is taken so that μ(B(x, r)) ≍ r^{d_{f}}, 0 < r ≤ 1. Then Λ_{p,q}^{β}(X) is defined as the space of all functions f ∈ L^{p}(X, μ) such that ā(β, f) := {a_{m}(β, f)}_{m=0}^{∞} ∈ l^{q}. Λ_{p,q}^{β}(X) is a Banach space with norm ∥f∥_{Λ_{p,q}^{β}(X)} := ∥f∥_{L^{p}} + ∥ā(β, f)∥_{l^{q}}. Different constants α, c_{0} give the same space with equivalent norm.

In the above theorem, it is known that F is described as Λ_{2,∞}^{d_{w}/2}(K) ([2,5]). Note that this theorem cannot be deduced just by combining the trace theorems on ℝ^{2} to K and to I. Since the traces of Dirichlet forms are canonically associated with jump processes on the trace sets, considering more general fractals is a natural problem from the probabilistic point of view.

To avoid notational complications, we consider here a concrete example. Let K be a n-dim. Sierpinski carpet, which is a subset of [0, 1]^{n}. Let μ be the normalized Hausdorff measure on K. Under some suitable conditions, there exists a nontrivial regular local self-similar Dirichlet form (E, F) on L^{2}(K, μ). It is known that F = Λ_{2,∞}^{d_{w}/2}(K), where d_{w}( > 2) is the walk dimension of (E, F). Let L = ([0, 1]^{n−1} × {0}) ∩ K. L is equipped with the normalized Hausdorff measure ν.

∗joint work with T. Kumagai (Kyoto Univ.)

Theorem. There exists a trace operator from F to Λ_{2,2}^{β}(L), where β = d_{w}/2−(d_{f} −d)/2, d_{f} = dim_{H} K, and d = dim_{H} L.

Since K is infinite ramified, the analysis becomes much harder.

References

[1] M. Hino and T. Kumagai, A trace theorem for Dirichlet forms on fractals, in preparation.

[2] A. Jonsson, Brownian motion on fractals and function spaces, Math. Z., 222 (1996), 496–504.

[3] A. Jonsson, A trace theorem for the Dirichlet form on the Sierpinski gasket, Math. Z., to appear.

[4] A. Jonsson and H. Wallin, Function spaces on subsets of ℝ^{n}, Mathematical Reports, Vol. 2, Part 1 (1984), Acad. Publ., Harwood.

[5] T. Kumagai, Brownian motion penetrating fractals -An application of the trace theorem of Besov spaces-, J. Func. Anal. 170 (2000), 69–92.

[6] H. Triebel, Fractals and Spectra, Monographs in Math., Vol. 91, Birkhäuser, Basel-Boston-Berlin, 1997.