Weight summability of solutions of the Sturm-Liouville equation (Q1283970)

Let \(G(x,t)\) be the Green function of the equation \[ -y''(x)+q(x)y(x)=f(x),\;\;x\in \mathbb{R}, \tag{1} \] with \(f(x)\in L_{p}(\mathbb{R})\), \(p\in [1,\infty]\) (\(L_{\infty}(\mathbb{R}):=C(\mathbb{R})\)) and \(1\leq q(x)\in L_{1}^{\text{loc}}(\mathbb{R})\). The operator \(G\) acting in \(L_{p}(\mathbb{R})\) is defined as follows \[ y(x)=(Gf)(x)\mathop{=}\limits^{\text{def}}\int_{-\infty}^{\infty}G(x,t)f(t)dt. \] The authors generalize their previous result by finding requirements for a weight \(r(x)\in L_{p}^{loc}(\mathbb{R})\) under which \[ | | r(\cdot)G| | <\infty. \] In the case \(r(x)\mathop{=}\limits^{a.e.}q(x)\) the problem is related to the problem of validity of the inequality \[ | | y''(x)| | _p +| | q(x)y(x)| | _p \leq c| | f(x)| | _p , \;\;f(x)\in L_p (\mathbb{R}), \] with an absolute constant \(c\) for the solutions \(y(x)\in L_p (\mathbb{R})\) to (1).