## Abstract

We consider the equation -y″(x)+q(x)y(x)=f(x), x ∈ ℝ, where f ∈ L _{1}(ℝ), 0 ≤ q ∈ L _{1} ^{loc}(ℝ), inf _{x∈ℝ} ∫ _{x-a} ^{x+a} q(t)dt>0 for some a>0. Under these conditions, (1) is correctly solvable in L _{1}(ℝ), i.e. (i) for any function f ∈ L _{1}(ℝ), there exists a unique solution of (1), y ∈ L _{1}(ℝ); (ii) there is an absolute constant c _{1} ∈ (0, ∞) such that the solution of (1), y ∈ L _{1}(ℝ), satisfies the inequality ∥y∥ _{1} ≤ c _{1}∥f∥ _{1} for all f ∈ L _{1}(ℝ). In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ L _{1} ^{loc}(ℝ), under which the solution of (1), y ∈ L _{1}(ℝ), satisfies the estimate ∥θy∥ _{1} ≤ c _{2}∥f∥ _{1} for all f ∈ L _{1}(ℝ), where c _{2} is some absolutely positive constant.

Original language | English |
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Pages (from-to) | 1175-1206 |

Number of pages | 32 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 141 |

Issue number | 6 |

DOIs | |

State | Published - 1 Dec 2011 |