On the iterated logarithmic Bloch space on the unit ball

We introduce a new space of analytic functions on the unit ball, the so-called iterated logarithmic Bloch space BlogkBlogk, and study some of its properties. Among other features, we study the boundedness and compactness of weighted composition operators on the unit disk, as well as of a recently introduced integral-type operator on the unit ball, with (i) domain the iterated logarithmic Bloch space BlogkBlogk, or the little iterated logarithmic Bloch space Blogk,0Blogk,0 and (ii) range the Bloch-type space BμBμ, or the little-Bloch-type space Bμ,0Bμ,0, or the weighted space Hμ∞, or the little weighted space Hμ,0∞.