  In this paper, we characterize the multiple operator integrals mappings which
are bounded on the Haagerup tensor product of spaces of compact operators. We
show that such maps are automatically completely bounded and prove that this is
equivalent to a certain factorization property of the symbol associated to the
operator integral mapping. This generalizes a result by
Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur
multipliers.
