Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913005624623104 |
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| author | Chang, Diwen Liu, Guanhua |
| author_facet | Chang, Diwen Liu, Guanhua |
| contents | In this paper, we focus on strongly local regular Dirichlet forms, especially those satisfying Morrey-type inequalities. We prove the equivalence between resistance estimates and heat kernel estimates in this case. Self-similar forms on fractals serve as a major application, where we construct a spatially inhomogeneous scaling function and characterize all the doubling self-similar measures. Further, on some special examples, the resistance conditions are reduced to some geometric conditions, on which a complete theory on self-similar Dirichlet spaces is established therein. In particular, we construct a concrete example on rotated triangle fractals, where the optimal heat kernel estimate is not related at all to the lower scaling exponent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03915 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions Chang, Diwen Liu, Guanhua Analysis of PDEs 35K08 In this paper, we focus on strongly local regular Dirichlet forms, especially those satisfying Morrey-type inequalities. We prove the equivalence between resistance estimates and heat kernel estimates in this case. Self-similar forms on fractals serve as a major application, where we construct a spatially inhomogeneous scaling function and characterize all the doubling self-similar measures. Further, on some special examples, the resistance conditions are reduced to some geometric conditions, on which a complete theory on self-similar Dirichlet spaces is established therein. In particular, we construct a concrete example on rotated triangle fractals, where the optimal heat kernel estimate is not related at all to the lower scaling exponent. |
| title | Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions |
| topic | Analysis of PDEs 35K08 |
| url | https://arxiv.org/abs/2604.03915 |