Endpoint boundedness of singular integrals: CMO space associated to Schrödinger operators
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| Format: | Preprint |
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2025
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| _version_ | 1866908334475444224 |
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| author | Han, Xueting Li, Ji Wu, Liangchuan |
| author_facet | Han, Xueting Li, Ji Wu, Liangchuan |
| contents | Let $ \mathcal{L} = -Δ+ V $ be a Schrödinger operator acting on $ L^2(\mathbb{R}^n) $, where the nonnegative potential $ V $ belongs to the reverse Hölder class $ RH_q $ for some $ q \geq n/2 $. This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $, consisting of functions of vanishing mean oscillation associated with $ \mathcal{L} $.
We establish the following main results: (i) the standard Hardy--Littlewood maximal operator is bounded on $\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; (ii) for each $ j = 1, \ldots, n$, the adjoint of the Riesz transform $ \partial_j \mathcal{L}^{-1/2} $ is bounded from $ C_0(\mathbb{R}^n) $ into $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; and (iii) the approximation to the identity generated by the Poisson and heat semigroups associated with $ \mathcal{L} $ characterizes $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ appropriately.
These results recover the classical analogues corresponding to the Laplacian as a special case. However, the presence of the potential $ V $ introduces substantial analytical challenges, necessitating tools beyond the scope of classical Calderón--Zygmund theory. Our approach leverages precise heat kernel estimates and the structural properties of $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ established by Song and the third author in [J. Geom. Anal. 32 (2022), no. 4, Paper No. 130, 37 pp]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Endpoint boundedness of singular integrals: CMO space associated to Schrödinger operators Han, Xueting Li, Ji Wu, Liangchuan Classical Analysis and ODEs 42B20, 42B25, 42B35 Let $ \mathcal{L} = -Δ+ V $ be a Schrödinger operator acting on $ L^2(\mathbb{R}^n) $, where the nonnegative potential $ V $ belongs to the reverse Hölder class $ RH_q $ for some $ q \geq n/2 $. This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $, consisting of functions of vanishing mean oscillation associated with $ \mathcal{L} $. We establish the following main results: (i) the standard Hardy--Littlewood maximal operator is bounded on $\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; (ii) for each $ j = 1, \ldots, n$, the adjoint of the Riesz transform $ \partial_j \mathcal{L}^{-1/2} $ is bounded from $ C_0(\mathbb{R}^n) $ into $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; and (iii) the approximation to the identity generated by the Poisson and heat semigroups associated with $ \mathcal{L} $ characterizes $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ appropriately. These results recover the classical analogues corresponding to the Laplacian as a special case. However, the presence of the potential $ V $ introduces substantial analytical challenges, necessitating tools beyond the scope of classical Calderón--Zygmund theory. Our approach leverages precise heat kernel estimates and the structural properties of $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ established by Song and the third author in [J. Geom. Anal. 32 (2022), no. 4, Paper No. 130, 37 pp]. |
| title | Endpoint boundedness of singular integrals: CMO space associated to Schrödinger operators |
| topic | Classical Analysis and ODEs 42B20, 42B25, 42B35 |
| url | https://arxiv.org/abs/2504.16827 |