Inner function in Dirichlet type spaces

Let X and Y   be two spaces of analytic functions in the unit disc DD with X⊆YX⊆Y. For an inner function θ, θ   is said to be (X,Y)(X,Y)-improving if fθ∈Xfθ∈X whenever f∈Xf∈X and fθ∈Yfθ∈Y. In this note, for a certain nonnegative and increasing function K   defined in [0;∞)[0;∞), we consider the space of Dirichlet type DKDK consisting of those functions f   analytic in DD such that ∫D|f′(z)|2K(1−|z|2)dA(z)<∞∫D|f′(z)|2K(1−|z|2)dA(z)<∞. We prove that an inner function belongs to DKDK if and only if it is (DK∩BMOA;BMOA)(DK∩BMOA;BMOA)-improving. We discuss also inner functions in DKDK as multipliers of DK∩BMOADK∩BMOA.