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Variation and oscillation inequalities for commutators in two-weight setting

Abstract This paper studies the two-weight estimates of variation and oscillation operators for commutators of singular integrals with weighted BMO{\mathrm{BMO}} functions. A new characterization of weighted BMO{\mathrm{BMO}} spaces via the boundedness of variation and oscillation operators for the iterated commutators of Calderón–Zygmund singular integrals in the two-weight setting is given.