On the exact asymptotics of small deviations of $L_2$-norm for some Gaussian random fields

In this paper we study the asymptotic behavior of the tail probability $\mathbf P(V^2<r)$ as $r\to 0$, where the sum $V^2$ is given by the formula $V^2=a^2 \sum_{i,j\ge 1} (i+\beta)^{-2c}(j+\delta)^{-2}\xi^2_{ij}$. Here $\{\xi_{ij}\}$ are independent standard Gaussian random variables, and $a>0$, $\beta >-1$, $\delta>-1$, $c>1/2$, $\ne 1$ are some constants. Thus, we study small deviations of the $L_2$-norm of certain two-parameter Gaussian random fields, that have the structure of a tensor product.