Differential and Integral Equations

Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem

N.A. Chernyavskaya and L.A. Shuster

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Abstract

We consider the equation \begin{equation} - y'(x)+q(x)y(x)=f(x),\quad x\in \mathbb R \tag*{(0.1)} \end{equation} where $f\in L_p(\mathbb R),$ $p\in[1,\infty]$ $(L_\infty(\mathbb R):=C(\mathbb R))$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ We assume that equation (0.1)} is correctly solvable in $L_p(\mathbb R).$ Let $y\in L_p(\mathbb R)$ be a solution of (0.1). In the present paper we find minimal requirements for the weight function $r\in L_p^{\rm loc}(\mathbb R)$ under which the following estimate holds: $$ \|ry\|_p\le c(p)\|f\|_p,\quad\forall f\in L_p(\mathbb R) $$ with an absolute constant $c(p)\in (0,\infty).$

Article information

Source
Differential Integral Equations, Volume 25, Number 5/6 (2012), 467-504.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012675

Mathematical Reviews number (MathSciNet)
MR2951737

Zentralblatt MATH identifier
1265.34135

Subjects
Primary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory 34A99: None of the above, but in this section

Citation

Chernyavskaya, N.A.; Shuster, L.A. Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem. Differential Integral Equations 25 (2012), no. 5/6, 467--504. https://projecteuclid.org/euclid.die/1356012675


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