Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

Abstract

AbstractConsider the equationwhereƒ∈Lp(ℝ),p∈ (1, ∞) andBy a solution of (*), we mean any functionyabsolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the spaceLp(ℝ), i.e.(1) for any function, there exists a unique solutiony∈Lp(ℝ) of (*);(2) there exists an absolute constantc1(p) > 0 such that the solutiony∈Lp(ℝ) of (*) satisfies the inequalityWe study the following problem on the strengthening estimate (**). Let a non-negative functionbe given. We have to find minimal additional restrictions forθunder which the following inequality holds:Here,yis a solution of (*) from the classLp(ℝ), andc2(p) is an absolute positive constant.