标题： 非负自伴算子相关的弱 Hardy 空间和球准范数函数空间上薛定谔群和虚幂算子的估计 显示英文标题

标题： 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

摘要： 设$(\mathbb{X},d,\mu)$是一个加倍度量测度空间，$L$是在$L^2(\mathbb{X})$上的一个非负自伴算子，满足 Davies-Gaffney 估计，$X(\mathbb{X})$是在$\mathbb{X}$上的一个球拟巴拿赫函数空间，满足一些温和的假设，且$p\in(0,\infty)$和$s_0\in(0,\min\{p,1\}]$。 在本文中，作者首先研究与$L$和$X(\mathbb{X})$相关的弱 Hardy 空间$WH_{X,L}(\mathbb{X})$，然后建立$WH_{X,L}(\mathbb{X})$的原子和小波分解。 作为应用，作者建立了分数次幂的Schrödinger群在$L$上的有界性估计，其中$WH_{X,L}(\mathbb{X})$:$$\left\|(I+L)^{-\beta/2}e^{i\tau L^{\gamma/2}}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$是一个常数，$0<\gamma\neq1$，$\beta\in[\gamma n(\frac{1}{s_0}-\frac{1}{2}),\infty)$，$r\in(0,1]$，$\tau\in \mathbb{R}$，以及$C>0$是一个常数。 此外，当$(\mathbb{X},d,\mu)$是一个 Ahlfors$n$正则的度量测度空间且$L$满足高斯上界估计时，作者还得到了$L$在$WH_{X,L}(\mathbb{X})$上的虚幂算子的有界性估计：$$\left\|L^{i\tau}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$其中$\alpha>n(\frac{1}{s_0}-\frac{1}{2})$，$r\in(\frac{n/s_0}{\alpha+n/2},1]$，$\tau\in \mathbb{R}$，和$C>0$是一个常数。 所有这些结果都具有广泛的普遍性，并且特别是当它们被应用于加权Lebesgue空间、Orlicz空间、可变Lebesgue空间和欧几里得空间设置时，这些结果也是新的。 显示英文摘要

摘要： Let $(\mathbb{X},d,\mu)$ 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 first study the weak Hardy space $WH_{X,L}(\mathbb{X})$ associated with $L$ and $X(\mathbb{X})$, and then establish the atomic and molecular decompositions of $WH_{X,L}(\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schr\"{o}dinger groups for fractional powers of $L$ on $WH_{X,L}(\mathbb{X})$: $$\left\|(I+L)^{-\beta/2}e^{i\tau L^{\gamma/2}}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $0<\gamma\neq1$, $\beta\in[\gamma n(\frac{1}{s_0}-\frac{1}{2}),\infty)$, $r\in(0,1]$, $\tau\in \mathbb{R}$, and $C>0$ is a constant. Moreover, when $(\mathbb{X},d,\mu)$ 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 $WH_{X,L}(\mathbb{X})$: $$\left\|L^{i\tau}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$ where $\alpha>n(\frac{1}{s_0}-\frac{1}{2})$, $r\in(\frac{n/s_0}{\alpha+n/2},1]$, $\tau\in \mathbb{R}$, and $C>0$ is a constant. All these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz spaces, variable Lebesgue spaces and Euclidean spaces setting, these results are also new.