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We consider the first order linear differential equation
\begin{align}\label{e1}
-y'(x)+q(x)y(x)=f(x), \ x \in R
\end{align}
where $f \in L_{p}(R)$, $p\in [1,\infty)$, $0\leq q \in
L_{1}^{loc}(R).$
Introduce the notation
\begin{align*}
q_{0}(a)=\underset{x\in R}{inf}\int\limits_{x-a}^{x+a}q(t)\,dt, \ \ a\in [0,\infty).
\end{align*}
For a continuous function $\theta$ such that $\theta(x)>0 $ for each $x \in R,$ let
$$L_{p,\theta }(R) = \{f\in L_{p}^{loc}(R):
\|f\|_{p,\theta}^{p}=\int\limits_{-\infty}^{\infty}|\theta(x)f(x)|^{p}\,dx0$ for some $ a\in
(0,\infty)$" is the minimal one to guarantee the fulfillment of
conditions (i) and (ii).
Here we consider the case when $\|q\|_{L_{1}(R)}=\infty, \ q_{0}(a)=0$ for all $
a\in [0,\infty)$ and find necessary and close to them sufficient
conditions for $\theta$ and $q$ under which (i) and (ii) are satisfied.
\begin{thebibliography}{99}
\bibitem{r1} Chernyavskaya N., {\it Conditions for correct solvability of a simplest singular boundary value problem}, Math. Nachr. {\bf 1} (2002), 5--18.
\bibitem{r2} Lukachev M., Shuster L., {\it On uniqueness of the solution of a linear differential equation without boundary conditions},
Functional Differential Equations {\bf 2} (2007), 337--346.
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