Pointwise regularity and irregularity of energy densities on $N$-dimensional Sierpinski gaskets
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911715011067904 |
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| author | Hino, Masanori Inui, Kanji Nitta, Kohei |
| author_facet | Hino, Masanori Inui, Kanji Nitta, Kohei |
| contents | 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 $γ_N=\log_2 \{(\sqrt{4N+5}+1)/(\sqrt{4N+5}-1)\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pointwise regularity and irregularity of energy densities on $N$-dimensional Sierpinski gaskets Hino, Masanori Inui, Kanji Nitta, Kohei Analysis of PDEs Probability Primary 31E05, Secondary 28A80, 31C25 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 $γ_N=\log_2 \{(\sqrt{4N+5}+1)/(\sqrt{4N+5}-1)\}$. |
| title | Pointwise regularity and irregularity of energy densities on $N$-dimensional Sierpinski gaskets |
| topic | Analysis of PDEs Probability Primary 31E05, Secondary 28A80, 31C25 |
| url | https://arxiv.org/abs/2602.06642 |