The problem is discussed of describing the weights $w$ on the unit circle for which the analytic and antianalytic subspaces of the corresponding weighted space $L^p(w)$ have nonzero intersection. In the special case of $p=2$ the problem is equivalent to a well-know problem about the exposed points in $H^1$. We show that the property in question is local, i.e., it depends on the local behavior of the weight $w$ at each point of the unit circle, and we obtain some necessary and sufficient condition in terms of Herglotz integrals.