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We consider a family of gradient Gaussian vector fields on $\Z^d$, where thecovariance operator is not translation invariant. A uniform finite rangedecomposition of the corresponding covariance operators is proven, i.e., thecovariance operator can be written as a sum of covariance operators whosekernels are supported within cubes of increasing diameter. An optimalregularity bound for the subcovariance operators is proven. We also obtainregularity bounds as we vary the coefficients defining the gradient Gaussianmeasures. This extends a result of S. Adams, R. Koteck\'y and S. M\"uller\cite{1202.1158}.

Finite range decomposition for a general class of elliptic operators