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 + q y M[y]=-y+qy on [ a , ∞ ) [a,\infty ) , a > − ∞ a>-\infty , where q ≥ 0 q\ge 0 and the mean value of q q computed over all subintervals of R \mathbb {R} of a fixed length is bounded away from zero, shares several standard “limit-point at ∞ \infty " properties of the L 2 L^2 case. We also show that there is a unique solution of M [ y ] = 0 M[y]=0 that is in all L p [ a , ∞ ) L^p[a, \infty ) , p = [ 1 , ∞ ] p=[1,\infty ] .