  This paper mainly dedicates to prove a plethora of weighted estimates on
Morrey spaces for bilinear fractional integral operators and their general
commutators with BMO functions of the form
$$B_{\alpha}(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha}}dy,\qquad
0<\alpha<n.$$ We also prove some maximal function control theorems for these
operators, that is, the weighted Morrey norm is bounded by the weighted Morrey
norm of a natural maximal operator when the weight belongs to $A_{\infty}$. As
a corollary, some new weighted estimates for the bilinear maximal function
associated to the bilinear Hilbert transform are obtained. Furthermore, we
formulate a bilinear version of Stein-Weiss inequality on Morrey spaces for
fractional integrals.
