Inner functions in the Möbius invariant Besov-type spaces

Abstract

Abstract An analytic function f in the unit disc $\mathbb{D}$ belongs to F ( p,q,s ), if is uniformly bounded for all a ∈ $\mathbb{D}$ . Here $g(z,a)=-\log|\varphi_a(z)|$ is the Green function of $\mathbb{D}$ , and φ a ( z )=( a−z )/(1−ā z ). It is shown that for 0 < γ < ∞ and | w |=1 the singular inner function exp(γ( z+w )/( z−w )) belongs to F ( p,q,s ), 0< s ≤1, if and only if $p\le q+\tfrac12(s+3)$ . Moreover, it is proved that, if 0< s <1, then an inner function belongs to the Möbius invariant Besov-type space $B^p_s=F(p,p-2,s)$ for some (equivalently for all) p > max{ s ,1− s } if and only if it is a Blaschke product whose zero sequence { z n } satisfies $\smash{\sup_{a\in\mathbb{D}}\sum_{n=1}^\infty(1-|\varphi_{a}(z_n)|^2)^s<\infty}$ .