On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910780572565504 |
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| author | Dziubański, Jacek |
| author_facet | Dziubański, Jacek |
| contents | Let $L=-Δ+ V(x)$ be a Schrödinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz transforms $$\big(\frac{\partial}{\partial x_1}L^{-1/2}, \frac{\partial}{\partial x_2}L^{-1/2},\dots,\frac{\partial}{\partial x_d}L^{-1/2}\Big).$$ The constant in the estimates does not depend on the potential $V$. We simultaneously provide a short proof of the weak type $(1,1)$ estimates for $\frac{\partial}{\partial x_j}L^{-1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators Dziubański, Jacek Functional Analysis 42B20 Let $L=-Δ+ V(x)$ be a Schrödinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz transforms $$\big(\frac{\partial}{\partial x_1}L^{-1/2}, \frac{\partial}{\partial x_2}L^{-1/2},\dots,\frac{\partial}{\partial x_d}L^{-1/2}\Big).$$ The constant in the estimates does not depend on the potential $V$. We simultaneously provide a short proof of the weak type $(1,1)$ estimates for $\frac{\partial}{\partial x_j}L^{-1/2}$. |
| title | On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators |
| topic | Functional Analysis 42B20 |
| url | https://arxiv.org/abs/2412.19922 |