We study the Aharonov-Bohm conductance oscillations of a constriction with an antidot in the fractional quantum Hall regime using a recently proposed composite-fermion Fermi liquid theory, and also using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. The predictions of the composite-fermion Fermi liquid theory are very similar to standard Fermi liquid theory and are consistent with recent experiments. In our chiral Luttinger liquid theory, which is valid in an experimentally realizable 'strongantidot-coupling' regime for bulk filling factors g = 1/q (q odd), the finite size of the antidot introduces a new temperature scale T 0 โก hv/ฯk B L, where v is the Fermi velocity and L is the circumference of the antidot edge state. Chiral Luttinger liquid theory predicts the low-temperature (T T 0 ) Aharonov-Bohm amplitude to vanish with temperature as T 2q-2 , in striking contrast to Fermi liquid theory (q = 1). Near T โ T 0 , there is a pronounced maximum in the amplitude, also in contrast to a Fermi liquid. At high temperatures (T T 0 ), however, we predict a new crossover to a T 2q-1 e -qT/T 0 temperature dependence, which is qualitatively similar to Fermi liquid behavior. We show how measurements in the strong-antidot-coupling regime, where transmission through the device is weak, should be able to distinguish between Fermi liquid and chiral Luttinger liquid behavior both at low and high temperatures and in the linear and nonlinear response regimes. Finally, we predict new mesoscopic edge-current oscillations, which are similar to persistent current oscillations in a mesoscopic ring, except that they are not reduced in amplitude by disorder. In the fractional regime, these 'chiral persistent currents' have a universal non-Fermi-liquid temperature dependence, and may be another ideal system to observe a chiral Luttinger liquid.