It is well known in the theory of bounded operators on $L_2(\mathbb{R}, \mu)$ that the operator $$ Tf(x) = \int_\mathbb{R} k(x,y) f(y) \: dy $$ is compact whenever $k(x,y) \in L_2(\mathbb{R}\times \mathbb{R})$. In my case I have the integral kernel, $$k(x,y) = e^{10x^2 - (x - \frac{1}{2}y)^2/2x^2}. $$ I wish to show that this $k(x,y) \in L_2(\mathbb{R} \times \mathbb{R})$. The measure, $\mu$, on my space is an intractable stationary distribution, but it decays on the order of $x^{-1/2}$ away from $0$. Is there a way to determine whether or not $T$ is indeed Hilbert-Schmidt for this $k$? What about if decay is of the order $x^{-1/2}e^{-hx}$?
I have attempted to use Schur's test, but I cannot find functions $p(x),q(x)$ which would give me a result here. Perhaps a clever choice could do the trick? I think that due to the slow rate of decay, $k$ probably not bounded.
Any help appreciated!