  In this paper, we consider localized integral operators whose kernels have
mild singularity near the diagonal and certain Holder regularity and decay off
the diagonal. Our model example is the Bessel potential operator ${\mathcal
J}_\gamma, \gamma>0$. We show that if such a localized integral operator has
stability on a weighted function space $L^p_w$ for some $p\in [1, \infty)$ and
Muckenhoupt $A_p$-weight $w$, then it has stability on weighted function spaces
$L^{p'}_{w'}$ for all $1\le p'<\infty$ and Muckenhoupt $A_{p'}$-weights $w'$.
