Geodesic
Real functions in weighted Hardy spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 27, Tome 262 (1999), pp. 138-146

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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. 
@article{ZNSL_1999_262_a5,
     author = {V. V. Kapustin},
     title = {Real functions in weighted {Hardy} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {138--146},
     publisher = {mathdoc},
     volume = {262},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_262_a5/}
}
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V. V. Kapustin. Real functions in weighted Hardy spaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 27, Tome 262 (1999), pp. 138-146. http://geodesic.mathdoc.fr/item/ZNSL_1999_262_a5/