Closure in logarithmic Hardy-Bloch norm of some analytic function spaces

Abstract

For 0<p,q<∞, the space of Hardy-Bloch type B(p,q) consists of those functions f which are analytic in the unit disk D such that (1−r)Mp(r,f′)∈Lq(dr/(1−r)). We note that B(p,p) is the Dirichlet space Dp−1p. For 0<p<∞, α∈R, a function f analytic in the unit disk D belongs to the logarithmic Hardy-Bloch space BHp,α if and only if sup0≤r<1⁡(1−r)(log⁡e1−r)αMp(r,f′)<∞. In this paper, we mainly characterize the closure of the Hardy-Bloch type space B(p,q) in logarithmic Hardy-Bloch space BHp,α when 1≤p<∞,1≤q<∞,α∈R. We also characterize the closure in the norm of BHp,α of the Dirichlet type spaces Dβp when 1≤p<∞,α∈R,β>−1.