Finite range decomposition for families of gradient Gaussian measures

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 the corresponding covariance operators, that is, the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters $ \sim L^k $. In addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient Gaussian measures.