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

Brown, R.

doi:10.1090/S0002-9939-04-07471-4

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

Proc. Amer. Math. Soc. 132 (2004), 2273-2280

DOI: https://doi.org/10.1090/S0002-9939-04-07471-4

Published electronically: March 25, 2004

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Abstract:

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 $L^p[a, \infty )$, $p=[1,\infty ]$.

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