Bulletin of the
Korean Mathematical Society BKMS

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: Primary 42B20, 42B25; Secondary 47B38

Supported by: The author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT $\&$ Future Planning(2015R1C1A1A02037331)