Hilbert matrix operator on Besov spaces

Abstract

We show that if 0 < ≤ ∞ 1, 1 < ≤ ∞ 1, then the Besov spaces H<inf>1+/=p</inf> ^p,q,1 are not mapped by the Hilbert matrix operator H into the Bloch space B. As a corollary, we have that the space VMOA is also not mapped by H into the Bloch space B. In [7], it is shown that if a function f(z) = ∑<inf>κ=0</inf>^∞ f(κ)z^κ, holomorphic in the unit disc, belongs to the logarithmically weighted Bergman space A² <inf>logα</inf>,α > 2, then ∑<inf>κ=0</inf>^∞ |f(κ)|/κ+1 < ∞. We show that this implication holds only when α > 1. In [7], it is also shown that if α > 3, then H maps A² <inf>logα</inf> into the Bergman space A². We improve this result by proving that H maps A<inf>2</inf> <inf>logα</inf> into A² when α > 2.