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As the classical $(p,q)$-Poincar\'e inequality is known to fail for $0 < p <1$, we introduce the notion of weighted multilinear Poincar\'e inequality as anatural alternative when $m$-fold products and $1/m < p$ are considered. Weprove such weighted multilinear Poincar\'e inequalities in the subellipticcontext associated to vector fields of H\"ormader type. We do so byestablishing multilinear representation formulas and weighted estimates formultilinear potential operators in spaces of homogeneous type.

Weighted multilinear Poincare inequalities for vector fields of Hormander type