Removable singularities for analytic functions in BMO and locally Lipschitz spaces

In this paper we study removable singularities for holomorphic functions such that sup_z|f⁽ⁿ⁾(z)|dist(z,)^s<. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. Dolzhenko (1963), Král (1976) and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on and be holomorphic on E, whereas we only assume that the functions belong to the function space on E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. Kaufman (1982) obtained some results for s=0.In this paper we obtain a number of examples with certain important properties. Similar examples have earlier been obtained for Hardy H^p classes and weighted Bergman spaces, mainly by the author. Because of the similarities in these three cases, an axiomatic approach is used to obtain some results that hold in all three cases with the same proofs.Supported by the Swedish Research Council and Gustaf Sigurd Magnusons fund of the Royal Swedish Academy of Sciences.Mathematics Subject Classification (2000):30B40, 30D45, 30D55, 46E15.