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

Abstract

We consider the equation

$ - (r(x)y'(x))' + q(x)y(x) = f(x),x \in \mathbb{R}$

((*))

where f ∈ L _p (ℝ), p ∈ (1,∞) and

$$r > 0,q \geqslant 0,\frac{1} {r} \in L_1^{loc} (\mathbb{R}),q \in L_1^{loc} (\mathbb{R})$$

,

$$\mathop {\lim }\limits_{|d| \to \infty } \int_{x - d}^x {\frac{{dt}} {{r(t)}}} \cdot \int_{x - d}^x {q(t)dt = \infty } $$

. 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,∞).