Admissible pair of spaces for not correctly solvable linear differential equations

TL;DR

This paper investigates the conditions under which certain weighted function spaces are suitable for solving a class of linear differential equations that are not correctly solvable in standard Lebesgue spaces, focusing on admissible pairs of spaces.

Contribution

It establishes criteria involving Otelbaev-type averages for the weight functions that make pairs of weighted and unweighted Lebesgue spaces admissible for these differential equations.

Findings

Identifies conditions on weight functions for admissibility.

Shows the role of Otelbaev-type averages in solvability.

Provides criteria for non-correctly solvable 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}}).