Weighted norm inequalities of various square functions and Volterra integral operators on the unit ball

In this paper, we investigate various square functions on the unit complex ball. We prove the weighted inequalities of the Lusin area integral associated with Poisson integral in terms of $A_p$ weights for all $1<p<\infty$; this gives an affirmative answer to an open question raised by Segovia and Wheeden. To that end, we establish the weighted inequalities for Littlewood-Paley type square functions. As an interesting application, we obtain the weighted inequalities of the Lusin area integral associated with Bergman gradient. In addition, we get an equivalent characterization of weighted Hardy spaces by means of the Lusin area integral in the context of holomorphic functions. We also obtain the weighted inequalities for Volterra integral operators. The key ingredients of our proof involve complex analysis, Calder\'on-Zygmund theory, the local mean oscillation technique and sparse domination.
Comment: 42 pages