This paper studies the two-weight estimates of variation and oscillation operators for commutators of singular integrals with weighted {\mathrm{BMO}} functions. A new characterization of weighted {\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.
In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new CMO(R n ) characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.
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