A limit-point criterion for a class of Sturm-Liouville operators defined in ${L^p}$ spaces

Using a recent result of Chernyavskaya and Shuster we show that the maximal operator determined by $M[y]=-y''+qy$ on $[a,\infty)$, $a>-\infty$, where $q\ge 0$ and the mean value of $q$ computed over all subintervals of $\mathbb{R}$ of a fixed length is bounded away from zero, shares several standard ``limit-point at $\infty$" properties of the $L^2$ case. We also show that there is a unique solution of $M[y]=0$ that is in all $L^p[a, \infty)$, $p=[1,\infty]$.