Abstract. The densities of invariant measures for Misiurewicz maps and LasotaYorke maps of class C are of class C r~l on certain intervals (forming the partition of an interval in case of Misiurewicz maps). For these maps the Perron-Frobenius operator has an unambiguous decomposition into the sum of projections onto eigenspaces (multiplied by the eigenvalues) and a remainder operator. The remainder operator has spectral radius less than one in certain spaces of smooth functions. Other useful references have maps with singularities where the derivative is equal to zero (for example the famous family of quadratic maps {ax(l -x)}). I recall only the papers closely related to the present research. M. Misiurewicz [6] has proved the existence and studied properties of absolutely continuous invariant measures for negative Schwarzian maps without sinks and such that the set of critical points is separated from the trajectory. W. Szlenk [10] has proved the existence of absolutely continuous measures for similar maps which are not necessarily negative Schwarzian.The third subject connected with this research is the question of smoothness of densities of invariant measures. This question for expanding maps was answered by R. Sacksteder [9] (unfortunately the paper contains an important mistake) and K. Krzyzewski [4] who proved that if an expanding map is of class C r then the density is of class C r~' .In this paper we construct the spaces C}~1 of functions of class C~l on intervals forming a partition with a weighted sup-norm.The Perron-Frobenius operator for Lasota-Yorke maps or Misiurewicz maps has an unambiguous decomposition into the sum of projections onto the eigenspaces and the remainder operator. The spectral radius of the remainder operator in the