Regularity properties of energy densities on the Sierpinski gasket

We investigate the regularity properties of energy densities associated with harmonic functions on the Sierpinski gasket with respect to the Kusuoka measure. While energy measures themselves have been extensively studied in the framework of analysis on fractals, the fine pointwise behavior of their densities has remained less well understood. We prove that the density of the energy measure of a nonconstant harmonic function is almost everywhere discontinuous, whereas it is Hölder continuous when restricted to each one-dimensional edge of the gasket.

💡 Research Summary

This paper studies the pointwise regularity of energy‑measure densities associated with harmonic functions on the N‑dimensional Sierpiński gasket (N ≥ 2) with respect to the Kusuoka measure ν. The authors extend earlier two‑dimensional results to arbitrary dimensions and provide a detailed analysis of both the pathological and the regular aspects of the Radon–Nikodym derivative dν_h/dν for a non‑constant harmonic function h.

The setting is the standard self‑similar Dirichlet form (E,F) on the gasket K, constructed via the usual iterated function system ψ_i(z) = (z + p_i)/2, where {p_i} are the vertices of an (N+1)‑simplex. Harmonic functions are uniquely determined by their values on the boundary vertices V₀, and the associated energy measures ν_f are defined by the standard integration by parts formula (2.3). The Kusuoka measure ν is the sum of the energy measures of the canonical harmonic functions h_k (k ∈ S), and every energy measure ν_f is absolutely continuous with respect to ν.

The main contributions are three theorems:

1. Theorem 2.1 (Almost‑everywhere discontinuity).
   For any non‑constant harmonic function h, there exists a Borel set A⊂K with ν(K\A)=0 such that any ν‑version of the Radon–Nikodym derivative dν_h/dν is discontinuous at every point of A. The proof uses the self‑similarity of ν_h (formula (2.4)), the spectral properties of the transition matrices A_k (eigenvalues 1, (N+1)/(N+3), 1/(N+3)), and asymptotic estimates for the cell measures ν(K_{w iⁿ}) and ν_h(K_{w iⁿ}). By examining the ratios ν_h(K_{w iⁿ})/ν(K_{w iⁿ}) along nested cells, the authors show that these ratios oscillate and fail to converge, which forces discontinuity on a full‑measure set.

2. Theorem 2.3 (Hölder continuity on edges).
   For any word w and distinct vertices i,j, the function δ_νh/δ_ν defined on the set ψ_w(p_i p_j)∩V* by the limit of the same ratios (taken along cells that approach a point from the two possible directions) is log₂(1+N⁻¹)‑Hölder continuous. The key technical device is Lemma 2.2, which guarantees that the two directional limits coincide, allowing a well‑defined δ on the dense set V*. The authors then study products of the matrices A_{w₁}…A_{w_n} acting on the (N−1)‑dimensional invariant subspace E_k. By constructing an invariant cone C⊂E_k and employing the Hilbert projective metric, they prove that the action of the matrix product on C is a strict contraction with factor (1+N⁻¹)⁻¹. This contraction yields a quantitative estimate on the variation of the ratios across neighboring cells, which translates directly into the stated Hölder exponent.

3. Theorem 2.4 (Full‑edge continuity for N=2,3).
   When N equals 2 or 3, the authors can extend δ_νh/δ_ν from V* to the entire edge ψ_w(p_i p_j). For each infinite word ω∈{i,j}^ℕ, the limit δ_h(ω)=lim_{n→∞} ν_h(K_{w