## Abstract

We consider the equation

−y′′(x)+q(x)y(x)=f(x),x∈R,

where f∈Lp(R), p∈[1,∞] (L∞(R):=C(R)) and

0≤q∈Lloc1(R);∃a>0:infx∈R∫x+ax−aq(t)dt>0,

(Condition (2) guarantees correct solvability of (1) in class Lp(R), p∈[1,∞].) Let y be a solution of (1) in class Lp(R), p∈[1,∞], and θ some non-negative and continuous function in R. We find minimal additional requirements to θ under which for a given p∈[1,∞] there exists an absolute positive constant c(p) such that the following inequality holds:

supx∈Rθ(x)|y(x)|≤c(p)∥f∥Lp(R) ∀f∈Lp(R).

−y′′(x)+q(x)y(x)=f(x),x∈R,

where f∈Lp(R), p∈[1,∞] (L∞(R):=C(R)) and

0≤q∈Lloc1(R);∃a>0:infx∈R∫x+ax−aq(t)dt>0,

(Condition (2) guarantees correct solvability of (1) in class Lp(R), p∈[1,∞].) Let y be a solution of (1) in class Lp(R), p∈[1,∞], and θ some non-negative and continuous function in R. We find minimal additional requirements to θ under which for a given p∈[1,∞] there exists an absolute positive constant c(p) such that the following inequality holds:

supx∈Rθ(x)|y(x)|≤c(p)∥f∥Lp(R) ∀f∈Lp(R).

Original language | American English |
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Pages (from-to) | 423-448 |

Number of pages | 26 |

Journal | Bolletino dell Unione Matematica Italiana |

Volume | 5 |

Issue number | 2 |

State | Published - 2012 |