  In this paper, the authors establish the two-weight boundedness of the local
fractional maximal operators and local fractional integrals on Gaussian measure
spaces associated with the local weights. More precisely, the authors first
obtain the two-weight weak-type estimate for the local-$a$ fractional maximal
operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with $1\leq
p\leq q<\infty$ under a condition of $(u,v)\in \bigcup_{b'>a}
A_{p,q,\alpha}^{b'}$, and then obtain the two-weight weak-type estimate for the
local fractional integrals. In addition, the authors obtain the two-weight
strong-type boundedness of the local fractional maximal operators under a
condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the
two-weight strong-type boundedness of the local fractional integrals. These
estimates are established by the radialization method and dyadic approach.
