Multipliers of dirichlet subspaces of the bloch space

Abstract

For 0 < p < ∞ we let Dpp-1 be the space of all functions f which are analytic in the unit disc D and satisfy R D (1-|Z|)p-1|f '(z)|pdA(z) < ∞. It is known that, whenever p ≠ q, the only multiplier from Dpp-1 to Dqq-1 is the trivial one. However, if X is a subspace of the Bloch space and 0 < p ≤ q < ∞, then X∩Dpp-1 ⊂ X∩ Dqq-1, a fact which implies that the space of multipliers M(Dpp-1∩X,Dqq-1∩X) is non-trivial. In this paper we study the spaces of multipliers M(Dpp-1∩X,Dqq-1∩X) (0 < p, q < ∞) for distinct classical subspaces X of the Bloch space. Specifically, we shall take X to be H∞, BMOA and the Bloch space B.