In the proof of Proposition A.11 on page 32 of this paper, the author takes the following step:
$$ \int_U \exp(-\|D_Gu\|^2_2/2 - \omega \|D u\|^2/2)du = \text{det}(I+\omega D_G^{-1} D^2 D_G^{-1})^{-1/2} \int_U \exp(-\|D_G w\|^2_2/2 )dw $$
which they explain is by substituting $(I+\omega D_G^{-1} D^2 D_G^{-1})^{1/2}u$ for $w$. To me this seems incorrect but given that it is central to the paper I am seeking a sanity check and likely missing something obvious. Note that the definition of $U$ is irrelevant to the question, $D_G, D$ are symmetric and positive definite matrices, and $\omega$ is some positive constant. Note that the exponent in the LHS can be written as
$$ -\|D_G u\|^2_2/2 - \omega \|D u\|^2/2 = -\frac{1}{2}u^T (D_G^2+ \omega D^2)u = -\frac{1}{2}\|(D_G^2 + \omega D^2)^{1/2} u\|^2_2, $$ and plugging in the proposed subsitution: $u = (I+\omega D_G^{-1} D^2 D_G^{-1})^{-1/2}w$ gives \begin{align*} -\frac{1}{2}\|(D_G^2 + \omega D^2)^{1/2} u\|^2_2 &= -\frac{1}{2}\|(D_G^2 + \omega D^2)^{1/2} (I+\omega D_G^{-1} D^2 D_G^{-1})^{-1/2} w\|^2_2\\ &= -\frac{1}{2}\|D_G^{1/2}(I + \omega D_G^{-1}D^2D_G^{-1})^{1/2}D_G^{1/2} (I+\omega D_G^{-1} D^2 D_G^{-1})^{-1/2} w\|^2_2\\ &\neq -\frac{1}{2}\|D_G w\|^2_2. \end{align*}
It seems to me that the correct substitution ought to have been: $D_G^{1/2}(I+\omega D_G^{-1} D^2 D_G^{-1})^{1/2} u=w$.