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Abstract
Suppose L= Δ+V is a Schrödinger operator on ℝ^{n} with a potential V belonging to certain reverse Hölder class RH_{σ} with σ ≥ n/ 2. The aim of this paper is to study the A_{p} weights associated to L, denoted by A_{p}^{L}, which is a larger class than the classical Muckenhoupt A_{p} weights. We first prove the quantitative A_{p}^{L} bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative A_{p,q}^{L} bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical A_{p,q} constant. However, since A_{p,q }⊂ A_{p,q}^{L}, the A_{p,q}^{L} constants are smaller than A_{p}_{,}_{q} constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the "exp–log" link between A_{p}^{L} and B M O_{L} (the BMO space associated with L), and show that for w ∈ A_{p}^{L}, log w is in B M O_{L}, and that the reverse is not true in general.
Original language  English 

Pages (fromto)  259283 
Number of pages  25 
Journal  Mathematische Zeitschrift 
Volume  293 
Issue number  12 
Early online date  14 Nov 2018 
DOIs  
Publication status  Published  Oct 2019 
Keywords
 Schrödinger operator
 Weighted inequalities
 Fractional integral operator
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Projects
 1 Finished

Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals
Duong, X., Ward, L., Li, J., Lacey, M., Pipher, J. & MQRES, M.
16/02/16 → 30/06/20
Project: Research