标题： 非负自伴算子和球准范数函数空间相关弱 Hardy 空间上的 Schrödinger 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，满足一些温和假设，且与$p\in(0,\infty)$和$s_0\in(0,\min\{p,1\}]$相关。 在本文中，作者研究与$L$和$X(\mathbb{X})$相关的弱 Hardy 空间$WH_{X,L}(\mathbb{X})$，然后给出$WH_{X,L}(\mathbb{X})$的原子和分子分解。 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，即在$WH_{X,L}(\mathbb{X})$上： $$\left\|(I+L)^{-\beta/2}e^{i\tau L^{\gamma/2}}f\right\|_{WH_{X,L}(\mathbb{X})}\leq C\left(1+|\tau|\right)^{n(\frac{1}{s_0}-\frac{r}{2})}\|f\|_{WH_{X,L}(\mathbb{X})},$$其中$0<\gamma\neq1$，$\beta\in[\gamma n(\frac{1}{s_0}-\frac{1}{2}),\infty)$， $r\in(0,1]$，$\tau\in \mathbb{R}$，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$WH_{X,L}(\mathbb{X})$上的虚数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。 这些结果对于强 Hardy 空间也是新颖的$H_{X,L}(\mathbb{X})$。此外，所有这些结果都具有广泛的普遍性，并且特别是当它们应用于加权 Lebesgue 空间、混合范数 Lebesgue 空间、Orlicz 空间、变指数 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 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 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. 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.