Comparison of Green kernels for elliptic operators on \((0,\infty)\) (Q1781873)

The aim of the paper is to compare Green's kernels of the following elliptic operators \[ L_{0}=\frac{1}{A}\frac{d}{dx}\left( A\frac{d}{dx}\right) -q \text{ and } L_{\lambda }=\frac{1}{A}\frac{d}{dx}\left( A\frac{d}{dx}\right) +\lambda g \frac{d}{dx}-q,\;\lambda \geq 0,\;x\in \left( 0,+\infty \right), \] under the assumptions: (i) \(A\) is a smooth function on \([0,+\infty )\) and positive on \(\left( 0,+\infty \right) \), (ii) \(q\) is continuous \ and nonnegative on \([0,+\infty )\) and (iii) \(g\) is continuous \ and nonnegative on \((0,+\infty )\). The authors provide necessary and sufficient conditions on the function \(g\) so that the \(L_{0}\)-Green's kernel and the \(L_{\lambda }\)-Green's kernel are comparable. They show that the set of such functions \(g\) is a convex cone.