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Weighted estimates for solutions of a Sturm–Liouville equation in the space L1(ℝ)

Published online by Cambridge University Press:  15 November 2011

N. A. Chernyavskaya ,
N. El-Natanov  and
L. A. Shuster
N. A. Chernyavskaya
Affiliation:
Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
N. El-Natanov
Affiliation:
Department of Mathematics, Ariel University Center of Samaria, Ariel 40700, Israel
L. A. Shuster
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel ([email protected])
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Abstract

We consider the equation

where

Under these conditions, (1) is correctly solvable in L1(ℝ), i.e.

(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);

(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),

In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimate

where c2 is some absolutely positive constant.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 6 , December 2011 , pp. 1175 - 1206
Copyright
Copyright © Royal Society of Edinburgh 2011

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