Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals

We study Schatten--von Neumann properties of multiple operator integrals with integrands in the Haagerup tensor product of $L^\infty$ spaces. We obtain sharp, best possible estimates. This allowed us to obtain sharp Schatten--von Neumann estimates in the case of {\it Haagerup-like} tensor products.