  We use precise asymptotic expansions for Jacobi functions
$\phi^{(\alpha,\beta)}_\lambda$ parameters $\alpha$, $\beta$ satisfying
$\alpha>1/2$, $\alpha>\beta>-1/2$, to generalizing classical H\"ormander-type
multiplier theorem for the spherical transform on a rank one Riemannian
symmetric space (by Clerc/Stein and Stanton/Tomas) to the framework of Jacobi
analysis. In particular, multiplier results for the spherical transform on
Damek--Ricci spaces are subsumed by this approach, and it yields multiplier
results for the hypergeometric `Heckman--Opdam transform' associated with a
rank one root system. We obtain near-optimal $L^p-L^q$ estimates for the
integral operator associated with the convolution kernel
$m_a:\lambda\mapsto(\lambda^2+\rho^2)^{-a/2}$, $a>0$.
