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We show that if $\Gamma$ is a discrete subgroup of the group of the isometries of ${ℍ}^k$, and if $\rho$ is a representation of $\Gamma$ into the group of the isometries of ${ℍ}^n$, then any $\rho$-equivariant map $F:{ℍ}^k\to {ℍ}^n$ extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable $\rho$-equivariant...