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.