Weighted norm inequalities for pluriharmonic conjugate functions (Q1604276)

The authors define pluriharmonic conjugate functions on the unit ball of \(\mathbb{C}^n\). Then they show that for a weight there exist weighted norm inequalities for pluriharmonic conjugate functions on \(L^p\) for \(1<p<\infty\) if and only if the weight satisfies the \(A_p\)-condition. The \(A_p\)-condition is related to the oscillation of function. As an application, they prove the equivalence of the weighted norm inequalities for the Cauchy integral and the \(A_p\)-condition of the weight. The authors show that there exist norm inequalities for pluriharmonic conjugate functions on BMO and on the nonisotropic Lipschitz spaces. As corollaries, in \(H^p\) for \(1<p<\infty\), the real part and the imaginary part of a function which fixes the origin are weighted norm equivalent. And in \(\text{BMOA}=H^2\cap \text{BMO}\), the real part and the imaginary part of a function which fixes the origin are norm equivalent.