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

TL;DR

This paper investigates the construction of Markov semigroup generators from a class of differential operators with measurable coefficients and singularities, extending the theory in $L^p$ and $C_ ablafty$ spaces.

Contribution

It establishes foundational results for realizing differential expressions with singular coefficients as operators generating Markov semigroups in $L^p$ and $C_ ablafty$ spaces.

Findings

Constructed operator realizations for differential expressions with singular coefficients.

Extended the theory of Markov semigroup generators to operators with critical-order singularities.

Provided conditions under which these operators generate Markov semigroups.

Abstract

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