The article deals with the correct solvability of some dierential equations in two-dimensional weighted Stepanov spaces. The solution is sought by methods of semigroup theory and related methods using fractional powers of operators. The results of the paper are important for the theory of dierential equations in Stepanov's spaces. Also they are important and form a basis for computational methods for dierential equations considered based on explicit integral representations for solutions. The results are obtained using ideas of S.G.Krein on fractional powers of operators in Banach spaces.