Non-divergence evolution operators modeled on Hörmander vector fields with Dini continuous coefficients
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915808280576000 |
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| author | Faini, Matteo |
| author_facet | Faini, Matteo |
| contents | In this paper we analyze operators H = a^{ij}(x,t) X_i X_j - d/dt (having adopted Einstein's convention on repeated indexes), where the X_i's are Hörmander vector fields generating a Carnot group and A = [a_{ij}] is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than Hölder continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight 2. As a consequence of the whole procedure, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-divergence evolution operators modeled on Hörmander vector fields with Dini continuous coefficients Faini, Matteo Analysis of PDEs 2020: 35K65, 35K08, 35K15 In this paper we analyze operators H = a^{ij}(x,t) X_i X_j - d/dt (having adopted Einstein's convention on repeated indexes), where the X_i's are Hörmander vector fields generating a Carnot group and A = [a_{ij}] is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than Hölder continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight 2. As a consequence of the whole procedure, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source. |
| title | Non-divergence evolution operators modeled on Hörmander vector fields with Dini continuous coefficients |
| topic | Analysis of PDEs 2020: 35K65, 35K08, 35K15 |
| url | https://arxiv.org/abs/2502.19073 |