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, \p, 1\]. In this article, the authors study the weak Hardy space WHₗ, ₋ (X) associated with L and X (X), and then give the atomic and molecular decompositions of WHₗ, ₋ (X). As applications, the authors establish the boundedness estimate of Schrödinger groups for fractional powers of L on WHₗ, ₋ (X): \| (I+L) ^-β/2e^iτL^{γ/2}f\|ₖ₇_ₗ, ₋ (X) C (1+|τ|) ^n (1{s₀-r2) }\|f\|ₖ₇_ₗ, ₋ (X), where 00 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 WHₗ, ₋ (X): \|L^iτf\|ₖ₇_ₗ, ₋ (X) C (1+|τ|) ^n (1{s₀-r2) }\|f\|ₖ₇_ₗ, ₋ (X), where α>n (1s₀-12), r (n/s₀α+n/2, 1], τ R, and C>0 is a constant. These results are also novelty for strong Hardy spaces Hₗ, ₋ (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.