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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.05333 |
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| _version_ | 1866913826841034752 |
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| author | Shi, Honglei Li, Pengtao Zhao, Kai |
| author_facet | Shi, Honglei Li, Pengtao Zhao, Kai |
| contents | Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative potential belonging to the reverse Hölder class. For $ α\in (0,1) $, we study regularity estimates of the fractional heat semigroups $ \{ exp (-tL^ {α} )\} _ { t > 0 }$, via the subordination formula and the fundamental solution of the associated uniformly parabolic equation $ \partial_t u + Lu = 0 $. This approach avoids the use of Fourier transforms and is applicable to second-order differential operators whose heat kernels satisfy Gaussian upper bounds. As an application, we characterize the Campanato-type space $Λ_{ L , γ} \left( \mathbb{R}^n \right)$ via the fractional heat semigroups $\{exp ( - t L ^ {α} ) \} _ { t > 0 } $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity estimates of fractional heat semigroups related with uniformly elliptic operators Shi, Honglei Li, Pengtao Zhao, Kai Functional Analysis Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative potential belonging to the reverse Hölder class. For $ α\in (0,1) $, we study regularity estimates of the fractional heat semigroups $ \{ exp (-tL^ {α} )\} _ { t > 0 }$, via the subordination formula and the fundamental solution of the associated uniformly parabolic equation $ \partial_t u + Lu = 0 $. This approach avoids the use of Fourier transforms and is applicable to second-order differential operators whose heat kernels satisfy Gaussian upper bounds. As an application, we characterize the Campanato-type space $Λ_{ L , γ} \left( \mathbb{R}^n \right)$ via the fractional heat semigroups $\{exp ( - t L ^ {α} ) \} _ { t > 0 } $. |
| title | Regularity estimates of fractional heat semigroups related with uniformly elliptic operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2505.05333 |