Methods of analysis of the condition for correct solvability in L _p(ℝ) of general Sturm-Liouville equations

We consider the equation (Formula presented.) where f ∈ L_p(ℝ), p ∈ (1,∞) and (Formula presented). In an earlier paper, we obtained a criterion for correct solvability of (*) in L_p(ℝ), p ∈ (1,∞). In this criterion, we use values of some auxiliary implicit functions in the coefficients r and q of equation (*). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function f(x) for x ∈ (a, b) through a function g(x) is sharp by order if c⁻¹|g(x)| ⩽ |f(x)| ⩽ c|g(x)|, x ∈ (a, b), c = const) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of (*) in L_p(ℝ), p ∈ (1,∞).