Singular integrals on $C^{1,α}$ intrinsic graphs in step 2 Carnot groups
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911132595257344 |
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| author | Chousionis, Vasileios Li, Sean Zhang, Lingxiao |
| author_facet | Chousionis, Vasileios Li, Sean Zhang, Lingxiao |
| contents | We study singular integral operators induced by Calderón-Zygmund kernels in any step-$2$ Carnot group $\mathbb{G}$. We show that if such an operator satisfies some natural cancellation conditions then it is $L^2$ bounded on all intrinsic graphs of $C^{1,α}$ functions over vertical hyperplanes that do not have rapid growth at $\infty$.
In particular, the result applies to the Riesz operator $\mathcal{R}$ induced by the kernel $$ \mathsf{R}(z)= \nabla_{\mathbb{G}} Γ(z), \quad z\in \mathbb{G}\backslash \{0\}, $$ the horizontal gradient of the fundamental solution of the sub-Laplacian. The $L^2$ boundedness of $\mathcal{R}$ is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that closed subsets of the intrinsic graphs mentioned above are non-removable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular integrals on $C^{1,α}$ intrinsic graphs in step 2 Carnot groups Chousionis, Vasileios Li, Sean Zhang, Lingxiao Classical Analysis and ODEs We study singular integral operators induced by Calderón-Zygmund kernels in any step-$2$ Carnot group $\mathbb{G}$. We show that if such an operator satisfies some natural cancellation conditions then it is $L^2$ bounded on all intrinsic graphs of $C^{1,α}$ functions over vertical hyperplanes that do not have rapid growth at $\infty$. In particular, the result applies to the Riesz operator $\mathcal{R}$ induced by the kernel $$ \mathsf{R}(z)= \nabla_{\mathbb{G}} Γ(z), \quad z\in \mathbb{G}\backslash \{0\}, $$ the horizontal gradient of the fundamental solution of the sub-Laplacian. The $L^2$ boundedness of $\mathcal{R}$ is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that closed subsets of the intrinsic graphs mentioned above are non-removable. |
| title | Singular integrals on $C^{1,α}$ intrinsic graphs in step 2 Carnot groups |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.09779 |