Weighted local Hardy spaces associated to Schrödinger operators

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In this paper, we characterize the weighted local Hardy spaces hᵖρ(ω) related to the critical radius function ρ and weights ω∈ A∞ρ,∞(Rⁿ) which locally behave as Muckenhoupt's weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with hᵖρ(ω). Furthermore, we establish boundedness in hᵖρ(ω) of quasi- Banach-valued sublinear operators. As their applications, we establish the equivalence of the weighted local Hardy space h¹ρ(ω) and the weighted Hardy space H¹ L(ω) associated to Schrödinger operators L with ω∈ A₁ρ,∞(Rⁿ)