  We consider a family of gradient Gaussian vector fields on the torus
$(\mathbb{Z}/L^N\mathbb{Z})^d$. Adams, Koteck\'{y}, M\"{u}ller and
independently Bauerschmidt established the existence of a uniform finite range
decomposition of the corresponding covariance operators, i.e., the covariance
can be written as a sum of covariance operators supported on increasing cubes
with diameter $L^k$. We improve this result and show that the decay behaviour
of the kernels in Fourier space can be controlled. Then we show the regularity
of the integration map that convolves functionals with the partial measures of
the finite range decomposition. In particular the new finite range
decomposition avoids the loss of regularity which arises in the renormalisation
group approach to anisotropic problems in statistical mechanics.
