A factorization theorem for the derivative of a function in $H^p$

Abstract:

We show that a function $G$ is the derivative of a function $f$ in the Hardy space $H^{p}$ of the unit disk $D$ for $0<p<\infty$ if and only if $G=F\Phi ^{\prime }$ where $F\in H^{p}$ and $\Phi \in BMOA$. Here, $F$ can be chosen to be non-vanishing, $||\Phi ||_{BMOA} \le 1$, and $||F||_{H^{p}} \le C||f||_{H^{p}}$. As an application, we characterize positive measures $\mu$ on the unit disk such that the operator $L_{\mu }g(\zeta )=\int _{D} g(z) {\frac {d\mu (z) }{(1-\zeta \bar {z})^{2}}}$ is bounded from the tent space $T^{p}_{\infty }$ to $H^{p}$, where ${\frac {1}{2}} <p<\infty$.