Boundary regularity for harmonic Hardy-Sobolev spaces

Two scales of harmonic Hardy-Sobolev spaces are introduced and their boundary regularity is studied. Both scales impose conditions on derivatives of harmonic functions in a fixed direction. In one case, they are required to have bounded /j-means, while in the other, they are required to have non-tangential maximal functions in L". The results include embedding in Lipschitz spaces, as well as into spaces of continuous functions and spaces of bounded and vanishing mean oscillation. In particular, real variable versions of the theorem of Privalov on analytic functions with absolutely continuous boundary values are proved