Admissible pair of spaces for not correctly solvable linear differential equations

We consider the differential equation begin{align}label{ab} -y(x)+q(x)y(x)=f(x), quad x in mathbb R, end{align} where $f in L_{p}(mathbb R)$, $pin [1,infty)$, and $0leq q in L_{1}^{rm loc}(mathbb R)$, $intlimits_{-infty}^{0}q(t),dt=intlimits_{0}^{infty}q(t),dt=infty,$ begin{align*} q_{0}(a)=inf_{xin mathbb R}int_{x-a}^{x+a}q(t),dt=0 quad{rm for ~ any }quad ain (0,infty). end{align*} Under these conditions, the equation ({rm ref{ab}}) is not correctly solvable in $L_{p}(mathbb R)$ for any $p in [1, infty) $. Let $q^{*}(x)$ be the Otelbaev-type average of the function $q(t), tin mathbb{R}$, at the point $t=x$; $theta(x)$ be a continuous positive function for $x in mathbb R$, and begin{align*} L_{p,theta }(mathbb R) = {fin L_{p}^{rm loc}(mathbb R):, int_{-infty}^{infty}|theta(x)f(x)|^{p},dx<infty }, end{align*} begin{align*} |f|_{L_{p,theta}(mathbb R)}=left(int_{-infty}^{infty}|theta(x)f(x)|^{p},dxright)^{1/p} end{align*} We show that if there exists a constant $cin [1, infty)$, such that the inequality $$c^{-1}q^{*}(x)leq theta(x)leq cq^{*}(x)$$ holds for all $x in mathbb{R}$, then under some additional conditions for $q$ the pair of spaces ${L_{p, theta}(mathbb R); L_{p}(mathbb R)}$ is admissible for the equation ({rm ref{ab}}).