Sharp integrability for Brownian motion in parabola-shaped regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains , . Together with some simple good- type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for by Bañuelos et al. (Ann. Probab. 29 (2001) 882) and for general by Li (Ann. Probab. 31 (2003) 1078). A sharp version of this result is also proved in higher dimensions.