Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Question

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in H_{0}^{1}(0,1)\quad$ compact? It's clear that this would be true if we would define $G$ on $L^{2}(0,1)$, but it is not at all obvious to me in the case of $H_{0}^{1}(0,1)$. As always, any help is greatly appreciated.