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 complex unit ball. We prove the weighted inequalities of the Lusin area integral associated with Poisson integral in terms of $A_p$ weights for all $1<p<\mathrm{\infty }$; this gives an affirmative answer to an open question raised by Segovia and Wheeden. 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.