Weighted estimates for solutions of a Sturm-Liouville equation in the space L 1(ℝ)

N. A. Chernyavskaya, N. El-Natanov, L. A. Shuster

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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∈ℝx-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 languageEnglish
Pages (from-to)1175-1206
Number of pages32
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume141
Issue number6
DOIs
StatePublished - 1 Dec 2011

Fingerprint

Dive into the research topics of 'Weighted estimates for solutions of a Sturm-Liouville equation in the space L <sub>1</sub>(ℝ)'. Together they form a unique fingerprint.

Cite this