A Continuous Right Inverse for Second Order Differential Operators with Variable Coefficients

Abstract

A continuous right inverse R for the second order linear differential operator \[ L = D_x^2 + p(x)D_x + q(x) \] with variable coefficients $p(x)$ and $q(x)$ is given. Moreover, there is a function $E(x,y)$ depending only on $p(x)$ and $q(x)$ such that \[ Rf(x) = \int_0^\pi {E(x,z)} \left( {\int_0^z {f(y)dy} } \right)dz \] for all f in the space on which L operates. Explicit formulas for $E(x,y)$ and moduli of continuity of R are given for various spaces of functions.