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

Let (X,d,μ) be a doubling metric measure space, L a non-negative self-adjoint operator on L²(X) satisfying the Davies-Gaffney estimate, and X(X) a ball quasi-Banach function space on X satisfying some mild assumptions with p∈(0,∞) and s₀∈(0,min,1\]. In this article, the authors study the weak Hardy space WHX,L(X) associated with L and X(X), and then give the atomic and molecular decompositions of WHX,L(X). As applications, the authors establish the boundedness estimate of Schrödinger groups for fractional powers of L on WHX,L(X): \|(I+L)-β/2e^iτLγ/2f\|_WHX,L(X)≤ C(1+|τ|)n(1/s₀-r/2)\|f\|_WHX,L(X), where 0<γ≠1, β∈[γn(1/s₀-1/2),∞), r∈(0,1], τ∈ R, and $C>0$ is a constant. Moreover, when (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 WHX,L(X): \|Liτf\|_WHX,L(X)≤ C(1+|τ|)n(1/s₀-r/2)\|f\|_WHX,L(X), where α>n(1/s₀-1/2), r∈(n/s₀/α+n/2,1], τ∈ R, and $C>0$ is a constant. These results are also novelty for strong Hardy spaces HX,L(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.