We study in this paper analytic Schur multipliers on C + 2 \mathbb {C}_{+}^{2} and D 2 \mathbb {D}^{2} , i.e. Schur multipliers on R 2 \mathbb {R}^{2} and T 2 \mathbb {T}^{2} that are boundary-value functions of functions analytic in C + 2 \mathbb {C}_{+}^{2} and D 2 \mathbb {D}^{2} . Such Schur multipliers are important when studying properties of functions of maximal dissipative operators and contractions under perturbation. We show that if the boundary-value function of a Schur multiplier has certain regularity properties, then it can be represented as an element of the Haagerup tensor product of spaces of analytic functions with similar regularity properties.