Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three
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arXiv
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| Formato: | Preprint |
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2023
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| _version_ | 1866914948957863936 |
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| author | Mizutani, Haruya Wan, Zijun Yao, Xiaohua |
| author_facet | Mizutani, Haruya Wan, Zijun Yao, Xiaohua |
| contents | This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,Δ^2)$ associated with fourth-order Schrödinger operators $H=Δ^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-μ}$ for some $μ>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<\infty$ under the condition that $μ>9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,Δ^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,Δ^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,Δ^2)$ and their dual operators $W_\pm(H,Δ^2)^*$ in the case where zero is a regular point and $μ>11$. These estimates depend critically on the singular integral theory of Calderón-Zygmund on a homogeneous space $(X,dω)$ with a doubling measure $dω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06768 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three Mizutani, Haruya Wan, Zijun Yao, Xiaohua Analysis of PDEs Mathematical Physics Classical Analysis and ODEs This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,Δ^2)$ associated with fourth-order Schrödinger operators $H=Δ^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-μ}$ for some $μ>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<\infty$ under the condition that $μ>9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,Δ^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,Δ^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,Δ^2)$ and their dual operators $W_\pm(H,Δ^2)^*$ in the case where zero is a regular point and $μ>11$. These estimates depend critically on the singular integral theory of Calderón-Zygmund on a homogeneous space $(X,dω)$ with a doubling measure $dω$. |
| title | Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2311.06768 |