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

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