Separation for differential operators and the $L^{p}$ spaces

Abstract:

The differential operator \[ My = {y^{(n)}} + {q_{n - 1}}{y^{(n - 1)}} + \cdots + {q_1}y’ + {q_0}y\] is said to be separated in ${L^p}(0,\infty )$ if $y \in {L^p}(0,\infty ),{y^{(n - 1)}}$ absolutely continuous, and $My \in {L^p}(0,\infty )$ imply that ${q_i}{y^{(i)}} \in {L^p}(0,\infty )$ for all $i = 0,1, \ldots ,n - 1$. As a special case of our main result we obtain: $M$ is separated in ${L^p}(0,\infty )$ if ${q_i} = {b_i} + {s_i}$ where ${b_i}$ is essentially bounded and ${s_i} \in {L^{pi}}(0,\infty )$ for some ${p_i} \geqslant p,i = 0,1, \ldots ,n - 1$. The case $n = 2,p = 2,{q_1} \equiv 0$ is due to Everitt-Giertz-Weidmann [2]. In the same paper these authors show that this result is best possible in the sense that for any $p$ satisfying $1 < p < 2$ a function $q$ exists in ${L^p}(0,\infty )$ such that $y'' + qy$ is not separated in ${L^2}(0,\infty )$.