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Let a family of gradient Gaussian vector fields on $ \mathbb{Z}^d $ be given.We show the existence of a uniform finite range decomposition of thecorresponding covariance operators, that is, the covariance operator can bewritten as a sum of covariance operators whose kernels are supported withincubes of diameters $ \sim L^k $. In addition we prove natural regularity forthe subcovariance operators and we obtain regularity bounds as we vary withinthe given family of gradient Gaussian measures.

Finite range decomposition for families of gradient Gaussian measures