Weighted Norm Inequalities for the Local Sharp Maximal Function

472 A.K. Lerner4. Proof of the Main ResultProof of Theorem 1. Since Mλ # |f |≤M# λf , we can assume that f ≥ 0. Supposealso that, additionally, f ∈ L ∞ ,ω ∈ L 1 ∩ L ∞ , and ω ≤ 2 m . Next, we use a standard“atomic decomposition” of ω based on the Calderón–Zygmund decomposition. For k ∈ Zwe write k ={x : M ω(x) > 2 k } as a disjoint union of dyadic cubes Q k jsuch that2 k 2 k }∣∣f − f Q kj∣ ∣∣ dx +∫R n fh −l dx∫M λ # nfdx+R fh n −l dx∫M λ # nfdx+R fh n −l dx .Since ‖h k ‖ 1 =‖ω‖ 1 for all k, the last integral can be estimated for any ε>0by∫∫R fh n −l dx ={x:f>ε}∫fh −l dx + fh −l dx{x:f ≤ε}≤ 2 n 2 −l ‖f ‖ ∞ |{x : f>ε}| + ε‖ω‖ 1 ,