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May/June 2012 Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem
N.A. Chernyavskaya, L.A. Shuster
Differential Integral Equations 25(5/6): 467-504 (May/June 2012).

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 $c(p)\in (0,\infty).$

Citation

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N.A. Chernyavskaya. L.A. Shuster. "Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem." Differential Integral Equations 25 (5/6) 467 - 504, May/June 2012.

Information

Published: May/June 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1265.34135
MathSciNet: MR2951737

Subjects:
Primary: 34A99, 34C10

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.25 • No. 5/6 • May/June 2012
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