  The $\gamma^{*}\rho^0\to\pi^0$ transition form factor is extracted from
recent result for the $\gamma^* \gamma^* \pi^0$ form factor obtained in the
extended hard-wall AdS/QCD model with a Chern-Simons term. In the large
momentum region, the form factor exhibits a $1/Q^4$ behavior, in accordance
with the perturbative QCD analysis, and also with the Light-Cone Sum Rule
(LCSR) result if the pion wave function exhibits the same endpoint behavior as
the asymptotic one. The appearance of this power behavior from the AdS side and
the LCSR approach seem to be rather similar: both of them come from the
{"}soft" contributions. Comparing the expressions for the form factor in both
sides, one can obtain the duality relation $z\propto \sqrt{u(1-u)}$, which is
compatible with one of the most important relations of the Light-Front
holography advocated by Brodsky and de Teramond. In the moderate $Q^2$ region,
the comparison of the numerical results from both approaches also supports a
asymptotic-like pion wave function, in accordance with previous studies for the
$\gamma^* \gamma^* \pi^0$ form factor. The form factor at zero momentum
transfer gives the $\gamma^{*}\rho^0\pi^0$ coupling constant, from which one
can determine the partial width for the $\rho^0(\omega)\to \pi^0 \gamma$ decay.
We also calculate the form factor in the time-like region, and study the
corresponding Dalitz decays $\rho^0(\omega)\to \pi^0 e^+e^-, \pi^0\mu^+\mu^-$.
Although all these results are obtained in the chiral limit, numerical
calculations with finite quark masses show that the corrections are extremely
small. Some of these calculations are repeated in the Hirn-Sanz model and
similar results are obtained.
