Distributions of Itô processes: estimates for the density and for conditional expectations of integral functionals

We obtain a priori estimates for the $L_2$-norms of solutions of parabolic Itô equations describing evolution of the distributions of solutions of ordinary Itô stochastic differential equations with random coefficients. In the case of nondegenerate equations, estimates for the $L_2$-norms of derivatives with respect to space variables are also obtained. As a consequence, we establish a generalisation of Itô's formula for functions that have only square-summable derivatives of the first and the second order (or even of the first order).