On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910380309086208 |
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| author | Kucharski, Maciej Wróbel, Błażej |
| author_facet | Kucharski, Maciej Wróbel, Błażej |
| contents | We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}Δ+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}Δ+V)^{-a}$ is bounded on $L^p(\mathbb{R}^d)$ with $1< p\leqslant 2$ whenever $a\leqslant 1/p.$ We demonstrate that the $L^{\infty}(\mathbb{R}^d)$ boundedness holds if $V$ satisfies an $a$-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on $V$ are also obtained on $L^{1}(\mathbb{R}^d).$ In particular our $L^{\infty}$ and $L^1$ results apply to non-negative potentials $V$ which globally have a power growth or an exponential growth. We also discuss a counterexample showing that the $L^{\infty}(\mathbb{R}^d)$ boundedness may fail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_15530 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators Kucharski, Maciej Wróbel, Błażej Functional Analysis Spectral Theory 47D08, 42B20, 42B37 We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}Δ+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}Δ+V)^{-a}$ is bounded on $L^p(\mathbb{R}^d)$ with $1< p\leqslant 2$ whenever $a\leqslant 1/p.$ We demonstrate that the $L^{\infty}(\mathbb{R}^d)$ boundedness holds if $V$ satisfies an $a$-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on $V$ are also obtained on $L^{1}(\mathbb{R}^d).$ In particular our $L^{\infty}$ and $L^1$ results apply to non-negative potentials $V$ which globally have a power growth or an exponential growth. We also discuss a counterexample showing that the $L^{\infty}(\mathbb{R}^d)$ boundedness may fail. |
| title | On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators |
| topic | Functional Analysis Spectral Theory 47D08, 42B20, 42B37 |
| url | https://arxiv.org/abs/2203.15530 |