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Let  L = − Δ + μ be the generalized Schrödinger operator on  ℝ d , d ≥ 3 , where  μ ≠ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new  BMO space associated to the generalized Schrödinger operator  L , BM O θ , L , which is bigger than the  BMO spaces related to the classical Schrödinger operators  A = − Δ + V , with  V a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to  L in  BM O θ , L also be proved.

End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator