A maximal function characterization of a class of Hardy spaces (Q1065311)

An analytic function \(f=u+i\tilde u\) in the upper half plane P belongs to \(H^ p\), \(0<p<\infty\), if and only if \[ \sup_{y>0}\sup_{v\in {\mathbb{R}}^ 1}\frac{1}{\pi}| f(x+iy)|^ p\frac{1}{1+\quad (v- x)^ 2}dx<\infty. \] When u is harmonic in P its maximal function \(u^*\) is defined by \(u^*(t)=\sup \{| u(x+iy)|:| x-t| <y\}\). The author proves necessary and sufficient results in connection with characterizing the \(H^ p\) spaces by the maximal function. The results give an extension of the results of \textit{D. L. Burkholder, R. F. Gundy} and \textit{M. L. Silverstein} [Trans. Am. Math. Soc. 157, 137-153 (1971; Zbl 0223.30048)].