  We prove that a probability solution of the stationary Kolmogorov equation
generated by a first order perturbation $v$ of the Ornstein--Uhlenbeck operator
$L$ possesses a highly integrable density with respect to the Gaussian measure
satisfying the non-perturbed equation provided that $v$ is sufficiently
integrable. More generally, a similar estimate is proved for solutions to
inequalities connected with Markov semigroup generators under the curvature
condition $CD(\theta,\infty)$. For perturbations from $L^p$ an analog of the
Log-Sobolev inequality is obtained. It is also proved in the Gaussian case that
the gradient of the density is integrable to all powers. We obtain
dimension-free bounds on the density and its gradient, which also covers the
infinite-dimensional case.
