Admissible pair of spaces for not correctly solvable linear differential equations

Abstract

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 $f\in L_p(\mathbb{R})$, $p\in [1,\infty )$, and $0\le q\in L_1^{\mathrm{loc}}(\mathbb{R})$, $\underset{-\infty }{\overset{0}{\int }}q(t)dt=\underset{0}{\overset{\infty }{\int }}q(t)dt=\infty ,$ \begin{align*} q_{0}(a)=\inf_{x\in \mathbb R}\int_{x-a}^{x+a}q(t)\,dt=0 \quad{\rm \ for ~ any }\quad a\in (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 $q^{\ast }(x)$ be the Otelbaev-type average of the function $q(t),t\in \mathbb{R}$, at the point $t=x$; $\theta (x)$ be a continuous positive function for $x\in \mathbb{R}$, and \begin{align*} L_{p,θ}(\mathbb R) = \{f\in L_{p}^{\rm loc}(\mathbb R):\, \int_{-\infty}^{\infty}|θ(x)f(x)|^{p}\,dx<\infty \}, \end{align*} \begin{align*} \|f\|_{L_{p,θ}(\mathbb R)}=\left(\int_{-\infty}^{\infty}|θ(x)f(x)|^{p}\,dx\right)^{1/p}\ \end{align*} We show that if there exists a constant $c\in [1,\infty )$, such that the inequality $$c^{-1}{q}^{\ast }(x)\le \theta (x)\le c{q}^{\ast }(x)$$ 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}}).