Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $f \in
L_{p}(\mathbb R)$, $p\in [1,\infty)$, 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}}).

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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol