  We study the discrete analogue of Stein-Weiss inequality and other variants
of fractional integrals on product space, we prove some
$\ell^p\rightarrow\ell^q$ bounds for these operators via iteration methods and
Fourier transform. In the last section, we involve a Dirichlet character in a
discrete fractional integral, then prove the regularity theorem by
Hardy-Littlewood circle method.
