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We obtain the basic results concerning the problem of constructing operatorrealizations of the formal differential expression $\nabla \cdot a \cdot \nabla- b \cdot \nabla$ with measurable matrix $a$ and vector field $b$ havingcritical-order singularities as the generators of Markov semigroups in $L^p$and $C_\infty$.

On the theory of the Kolmogorov operator in the spaces $L^p$ and $C_\infty.$ I