Interior Estimates of Solutions of Linear Differential Inequalities

Abstract

For the solutions of the linear differential inequality $\mathscr {L}(u)\geq 0$, where $\mathscr {L}$ is a linear differential operator of order $l$ defined on functions of one variable, we establish estimates of the form $\|u; W^l (J^\delta )\|\leq C(\delta )\|u; L(J)\|$, where $J=[a,b]\subset \mathbb {R}$, $0<3\delta <b-a$, $J^\delta =[a+\delta , b-\delta ]$, $W^l (J^\delta )$ is the Sobolev space of $l$ times differentiable functions, $L(J)$ is the Lebesgue space of integrable functions, and the constant $C(\delta )$ is independent of the choice of the function $u$.