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 $0\leq q \in L_{1}^{\rm loc}(\mathbb R)$, $\int\limits_{-\infty}^{0}q(t)\,dt=\int\limits_{0}^{\infty}q(t)\,dt=\infty,$ \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 $q^{*}(x)$ 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,\theta }(\mathbb R) = \{f\in 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}\,dx\right)^{1/p}\ \end{align*} We show that if there exists a constant $c\in [1, \infty)$, such that the inequality $$c^{-1}q^{*}(x)\leq \theta(x)\leq cq^{*}(x)$$ 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}}).