  We consider a one dimensional mesoscopic capacitor in the presence of strong
electron interactions and compute its admittance in order to probe the
universal nature of the relaxation resistance. We use a combination of
perturbation theory, renormalization group arguments, and quantum Monte Carlo
calculation to treat the whole parameter range of dot-lead coupling. The
relaxation resistance is universal even in the presence of strong Coulomb
blockade when the interactions in the wire are sufficiently weak. We predict
and observe a quantum phase transition to an incoherent regime for a Luttinger
parameter $K<1/2$. Results could be tested using a quantum dot coupled to an
edge state in the fractional quantum Hall effect.
