On distribution of quadratic forms in Gaussian random variables

Two-sided bounds are constructed for a density function $p(u,a)$ of a random variable $|Y-a|^2 $, where $Y$ is a Gaussian random element in a Hilbert space with zero mean. The estimates are sharp in the sense that starting from large enough $u$ the ratio of upper bound to lower bound equals 8 and does not depend on any parameters of a distribution of $|Y-a|^2$. The estimates imply two-sided bounds for probabilities $\mathbf{P}(|Y-a|>r)$