Operator of Logarithmic Weight Bloch Type Space and Weakly Compact Composition Operator

Abstract

We deal with the integral–type operator from Bloch space, logarithmic Bloch space and Dirichlet space to the Bloch-type space on the unit ball. The norm of operators from logarithmic Bloch type spaces to weighted- type spaces are considered. We give the composition of Bloch with bounded analytic and inner functions, symmetric measures and biBloch mapping. The Bloch-to-BMOA compositions on complex balls, reverse estimates in logarithmic and weight Bloch spaces and quadratic integrals are established. The composition operators from Bloch type spaces to Hardy and Besov spaces are discussed with the compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces are found.