Derivatives of Hardy functions

Abstract:

Let $B$ be the open unit ball of ${C^n}$, and set $S = \partial B$. It is shown that if $\varphi \in {L^p}\left ( S \right ),\varphi > 0$, is a lower semicontinuous function on $S$ and $1/q > 1 + 1/p$, then, for a given $\varepsilon > 0$, there exists a function $f \in {H^p}\left ( B \right )$ with $f\left ( 0 \right ) = 0$ such that $\left | {{f^ * }} \right | = \varphi$ almost everywhere on $S$ and $\int _B {{{\left | {\nabla f} \right |}^q}dV < \varepsilon }$ where $V$ denotes the normalized volume measure on $B$.