Bloch functions with wild boundary behavior in CN${\mathbb {C}}^N$

Abstract We prove the existence of functions in the Bloch space of the unit ball of with the property that, given any measurable function on the unit sphere , there exists a sequence , , converging to 1, such that for every , The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.