  We introduce a notion of holomorphic stability for pairs of function spaces
on a planar domain $\Omega$. In the case of the open unit disk $\Omega =
\mathbb{D}$ equipped with a radial measure $\mu$, by establishing
Bourgain-Brezis type inequalities, we show that the pair \[(B^2(\mathbb{D},
\mu), h^1(\mathbb{D}))\] of weighted harmonic Bergman space and harmonic Hardy
space is holomorphically stable if and only if $\mu$ is a $(1,2)$-Carleson
measure. With some extra efforts, we also obtain an analogous result for the
upper half plane equipped with horizontal-translation invariant measures.
