• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.205-215
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• 2021

In [1], Beznosova proved that the bound on the norm of the dyadic paraproduct with b ∈ BMO in the weighted Lebesgue space L2(w) depends linearly on the Ad2 characteristic of the weight w and extrapolated the result to the Lp(w) case. In this paper, we provide the weighted norm estimates of the dyadic paraproduct πb with b ∈ VMO and reduce the dependence of the Ad2 characteristic to 1/2 by using the property that for b ∈ VMO its mean oscillations are vanishing in certain cases. Using this result we also reduce the quadratic bound for the commutators of the Calderón-Zygmund operator [b, T] to 3/2.