Dunford-Pettis composition operators on $H^{1}$ in several variables

A bounded composition operator $C_{\phi }$ on $H^{1}(B)$, where $B$ is the unit ball in ${\mathbb{C}}^{n}$, is Dunford-Pettis if and only if the radial limit function $\phi ^{*}$ of $\phi$ takes values on the unit sphere $S$ only on a set of surface measure zero. A similar theorem holds on bounded strongly pseudoconvex domains with smooth boundary.