On summability with a weight of a solution of the Sturm-Liouville equation

M. O. Otelbaev

Matematičeskie zametki, Tome 16 (1974) no. 6, pp. 969-980

Résumé

We study problems of summability with a weight of a solution of the Sturm–Liouville equation $$ -y'+q(x)y=f,\quad x\in J=(-\infty,\infty). $$ with bounded potential $q(x)$, satisfying the conditions \begin{gather*} \inf_{x\in J}q(x)\ge-\mu+1,\quad\sup_{|x-y|\le2}\frac{q(x)+\mu}{q(y)+\mu}<+\infty, \\ \sup_{|x-y|\le2}\{|x-y|^{-\alpha}|q(x)|^{-\alpha}\exp(-r|x-y|\sqrt{q(x)+\lambda})|q(x)-q(y)|\}<+\infty, \end{gather*} where $\alpha\in(0,1]$, $r\in[0,1)$, $2-2a+\alpha>0$, $a\ge0$, $\mu\ge0$. Our main result is the following: let $q(x)$ satisfy the conditions given above and let l$(x)$ be a nonnegative function such that $$ C(|x|^C+1)\ge l(x)\ge C^{-1}(|x|^C+1)^{-1},\quad\sup_{|x-y|\le2}\frac{l(x)}{l(y)}<+\infty, $$ then if $-y''+q(x)y=f$ и $y(x)l(x),~f(x)l(x)\in L_p(J)$ ($1\le p<\infty$), it follows that \begin{gather*} y''l(x),\quad q(x)l(x)y(x), \\ (q(x)+\mu)^{1/2}y'(x)l(x)\in L_p(J). \end{gather*}