Weighted norm estimates for the dyadic paraproduct with VMO function
Bull. Korean Math. Soc.
Published online November 2, 2020
Daewon Chung
Keimyung University
Abstract : In \cite{Be}, Beznosova proved that the bound on the norm of the dyadic paraproduct with $b\in $BMO in the weighted Lebesgue space $L^2(w)$ depends linearly on the $A_2^d$ characteristic of the weight $w$ and extrapolated the result to the $L^p(w)$ case. In this paper, we provide the weighted norm estimates of the dyadic paraproduct $\pi_b$ with $b\in$ VMO and reduce the dependence of the $A_2^d$ characteristic to $1/2$ by using the property that for $b\in$ VMO its mean oscillations are vanishing in certain cases. Using this result we also reduce the quadratic bound for the commutators of the Calder\'{o}n-Zygmund operator $[b,T]$ to $3/2$.
Keywords : Weighted norm estimate, Dyadic paraproduct, $A_2$-weights, Carleson sequence
MSC numbers : 42B20, 42B25
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