Estimates of Solutions of Parabolic Equations for Measures

Estimates of solutions of parabolic equations for measures are studied. A parabolic equation is considered that depends on the coefficients that are locally integrable with respect to the measure and the equation is understood in the sense of the integral identity. If a finite Borel measure satisfies the inequality, the measure has a certain density with respect to Lebesgue measure. The results also show that the measure satisfies inequality and a the new coordinates are introduced instead of the old coordinates. For a probability measure that is a solution of the Cauchy problem, the measure has density with respect to Lebesgue measure.