Estimates for Schrödinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces
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2025
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| author | Liu, Xiong Wang, Wenhua |
| author_facet | Liu, Xiong Wang, Wenhua |
| contents | Let $(\mathbb{X},d,μ)$ be a doubling metric measure space, $L$ a non-negative self-adjoint operator on $L^2(\mathbb{X})$ satisfying the Davies-Gaffney estimate, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$ satisfying some mild assumptions with $p\in(0,\infty)$ and $s_0\in(0,\min\{p,1\}]$. In this article, the authors study the weak Hardy space $WH_{X,L}(\mathbb{X})$ associated with $L$ and $X(\mathbb{X})$, and then give the atomic and molecular decompositions of $WH_{X,L}(\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schrödinger groups for fractional powers of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|(I+L)^{-β/2}e^{iτL^{γ/2}}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|τ|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $0<γ\neq1$, $β\in[γn(\frac{1}{s_0}-\frac{1}{2}),\infty)$, $r\in(0,1]$, $τ\in \mathbb{R}$, and $C>0$ is a constant. Moreover, when $(\mathbb{X},d,μ)$ is an Ahlfors $n$-regular metric measure space and $L$ satisfies the Gaussian upper bound estimate, the authors also obtain the boundedness estimate of imaginary power operators of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|L^{iτ}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|τ|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $α>n(\frac{1}{s_0}-\frac{1}{2})$, $r\in(\frac{n/s_0}{α+n/2},1]$, $τ\in \mathbb{R}$, and $C>0$ is a constant. These results are also novelty for strong Hardy spaces $H_{X,L}(\mathbb{X})$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to weighted Lebesgue spaces, mixed-norm Lebesgue spaces, Orlicz spaces, variable Lebesgue spaces and Euclidean spaces setting, these results are also new. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_13913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimates for Schrödinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces Liu, Xiong Wang, Wenhua Classical Analysis and ODEs Primary 42B35, Secondary 42B30, 42B25 Let $(\mathbb{X},d,μ)$ be a doubling metric measure space, $L$ a non-negative self-adjoint operator on $L^2(\mathbb{X})$ satisfying the Davies-Gaffney estimate, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$ satisfying some mild assumptions with $p\in(0,\infty)$ and $s_0\in(0,\min\{p,1\}]$. In this article, the authors study the weak Hardy space $WH_{X,L}(\mathbb{X})$ associated with $L$ and $X(\mathbb{X})$, and then give the atomic and molecular decompositions of $WH_{X,L}(\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schrödinger groups for fractional powers of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|(I+L)^{-β/2}e^{iτL^{γ/2}}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|τ|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $0<γ\neq1$, $β\in[γn(\frac{1}{s_0}-\frac{1}{2}),\infty)$, $r\in(0,1]$, $τ\in \mathbb{R}$, and $C>0$ is a constant. Moreover, when $(\mathbb{X},d,μ)$ is an Ahlfors $n$-regular metric measure space and $L$ satisfies the Gaussian upper bound estimate, the authors also obtain the boundedness estimate of imaginary power operators of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|L^{iτ}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|τ|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $α>n(\frac{1}{s_0}-\frac{1}{2})$, $r\in(\frac{n/s_0}{α+n/2},1]$, $τ\in \mathbb{R}$, and $C>0$ is a constant. These results are also novelty for strong Hardy spaces $H_{X,L}(\mathbb{X})$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to weighted Lebesgue spaces, mixed-norm Lebesgue spaces, Orlicz spaces, variable Lebesgue spaces and Euclidean spaces setting, these results are also new. |
| title | Estimates for Schrödinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces |
| topic | Classical Analysis and ODEs Primary 42B35, Secondary 42B30, 42B25 |
| url | https://arxiv.org/abs/2508.13913 |