L^q densities for measures associated with parabolic IFS with overlaps

We study parabolic iterated function systems (IFS) with overlaps on the real line and measures associated with them. A Borel probability measure μ on the coding space projects into a measure ν on the limit set of the IFS. We consider families of IFS satisfying a transversality condition. In [SSU2] sufficient conditions were found for the measure ν to be absolutely continuous for Lebesgue-a.e. parameter value. Here we investigate when ν has a density in L^q (ℝ) for q > 1. A necessary condition is that the q-dimension of μ (computed with respect to a certain metric associated with the IFS) is greater or equal to one. We prove that this is sharp for 1 < q ≤ 2 in the following sense: if μ is a Gibbs measure with a Hölder continuous potential, then ν has a density in L^q (ℝ) for Lebesgue-a.e. parameter value such that the q-dimension of μ is greater than one. This result is applied to a family of random continued fractions studied by R. Lyons.