Geodesic
Carleson measures and uniformly perfect sets
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 92-103

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We show that the description of Carleson measures on the Bergman spase of analytic functions on a finitely connected domain $G$ with the power weight is the same one as in the unit disk iff the complement $\overline{\mathbb C}\setminus G$ be an unbounded set without isolated points. In general case the complement of such domain $G$ have to be a uniformly perfect set. 
@article{ZNSL_1998_255_a5,
     author = {V. L. Oleinik},
     title = {Carleson measures and uniformly perfect sets},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {92--103},
     publisher = {mathdoc},
     volume = {255},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_255_a5/}
}
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V. L. Oleinik. Carleson measures and uniformly perfect sets. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 92-103. http://geodesic.mathdoc.fr/item/ZNSL_1998_255_a5/