  We study the sharp order of integrability of the exit position of Brownian
motion from the planar domains ${\cal P}_\alpha = \{(x,y)\in \bR\times
\bR\colon x> 0, |y| < Ax^{\alpha}\}$, $0<\alpha<1$. Together with some simple
good-$\lambda$ type arguments, this implies the order of integrability for the
exit time of these domains; a result first proved for $\alpha =1/2$ by
Ba\~nuelos, DeBlassie and Smits \cite{ba} and for general $\alpha$ by Li
\cite{li}. A sharp version of this result is also proved in higher dimensions.
