Abstract
We consider the equation
where f ∈ L p (ℝ), p ∈ (1,∞) and
,
. 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,∞).
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Chernyavskaya, N.A., Shuster, L.A. Methods of analysis of the condition for correct solvability in L p (ℝ) of general Sturm-Liouville equations. Czech Math J 64, 1067–1098 (2014). https://doi.org/10.1007/s10587-014-0154-1
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DOI: https://doi.org/10.1007/s10587-014-0154-1