  We review the recent developments on the light-flavor resonances in the
$U(3)$ chiral effective filed theory. The spectral function sum rules and the
semilocal duality in the scattering, which will be focus of this note, can
provide us interesting and useful theoretical objects to bridge the hadron
resonances in the intermediate energy region and the QCD behaviors in the
asymptotic region. First the calculations of the meson-meson scattering
amplitudes and factor factors are elaborated. The scalar spectral functions are
then calculated in terms of the unitarized scalar form factors. The scalar and
pseudoscalar spectral function sum rules in our study are found to be
consistent with the asymptotic behavior of QCD in the chiral limit. The
semilocal duality is found to be generally well satisfied, indicating the
necessary cancellations of different contributions from different resonances
indeed happen in the scattering amplitudes. The $N_C$ evolutions of the
resonance poles, the ratios to quantify the semilocal duality and the spectral
function integrals are also paid special attention to in this note.
