Pointwise regularity and irregularity of energy densities on $N$-dimensional Sierpinski gaskets

Abstract

We study the pointwise regularity of energy densities associated with harmonic functions on the $N$-dimensional Sierpinski gasket $(N\ge 2)$ with respect to the Kusuoka measure. For any nonconstant harmonic function, we prove that every Borel representative of the density is discontinuous at every point of a set of full Kusuoka measure. In sharp contrast, on each one-dimensional edge of the gasket -- itself a set of zero Kusuoka measure -- the density admits a canonical pointwise version, which is $γ_N$-Hölder continuous on that edge with the explicit and optimal exponent $γ_N=\log_2 \{(\sqrt{4N+5}+1)/(\sqrt{4N+5}-1)\}$.