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

Abstract

Let $(\mathbb{X},d,\hat{I}¼)$ 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 $W{H}_{X,L}(\mathbb{X})$ associated with $L$ and $X(\mathbb{X})$, and then give the atomic and molecular decompositions of $W{H}_{X,L}(\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schrödinger groups for fractional powers of $L$ on $W{H}_{X,L}(\mathbb{X})$: $${\Vert (I+L{)}^{-{\hat{I}}^2/2}{e}^{i\ddot{I}„L^{{\hat{I}}^3/2}}f\Vert }_{W{H}_{X,L}(\mathbb{X})}\le C{\left(1+|\ddot{I}„|\right)}^{n(\frac{1}{s_0}-\frac{r}{2})}\Vert f{\Vert }_{W{H}_{X,L}(\mathbb{X})},$$ where $0<{\hat{I}}^3\ne 1$, ${\hat{I}}^2\in [{\hat{I}}^{3}n(\frac{1}{s_0}-\frac{1}{2}),\infty )$, $r\in (0,1]$, $\ddot{I}„\in \mathbb{R}$, and $C>0$ is a constant. Moreover, when $(\mathbb{X},d,\hat{I}¼)$ 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 $W{H}_{X,L}(\mathbb{X})$: $${\Vert L^{i\ddot{I}„}f\Vert }_{W{H}_{X,L}(\mathbb{X})}\le C{\left(1+|\ddot{I}„|\right)}^{n(\frac{1}{s_0}-\frac{r}{2})}\Vert f{\Vert }_{W{H}_{X,L}(\mathbb{X})},$$ where $\hat{I}\pm >n(\frac{1}{s_0}-\frac{1}{2})$, $r\in (\frac{n/s_0}{\hat{I}\pm +n/2},1]$, $\ddot{I}„\in \mathbb{R}$, 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.

37 pages