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We give a new approach, inspired by H\"ormander's $L^2$-method, to weightedvariance inequalities which extend results obtained by Bobkov and Ledoux. Itprovides in particular a local proof of the dimensional functional forms of theBrunn-Minkowski inequalities. We also present several applications of thesevariance inequalities, including reverse H\"older inequalities for convexfunctions, weighted Brascamp-Lieb inequalities and sharp weighted Poincar\'{e}inequalities for generalized Cauchy measures.

Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Pr\'ekopa's theorem