Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation

Chernyavskaya, N. A. ; Shuster, L. A.

Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 423-448

Résumé

We consider the equation \begin{equation*} \tag{1} -y''(x) + q(x)y(x) = f(x), \qquad x \in \mathbb{R}, \end{equation*} where $f \in L_{p}(\mathbb{R})$, $p \in [1,\infty]$ ($L_{\infty}(\mathbb{R}) := C(\mathbb{R})$) and \begin{equation*} \tag{2} 0 \leq q \in L_{1}^{\text{loc}}(\mathbb{R}); \qquad \exists a > 0 : \inf_{x \in \mathbb{R}} \int_{x-a}^{x+a} q(t) \, dt > 0, \end{equation*} (Condition (2) guarantees correct solvability of (1) in class $L_{p}(\mathbb{R})$, $p \in [1,\infty]$.) Let $y$ be a solution of (1) in class $L_{p}(\mathbb{R})$, $p \in [1,\infty]$, and $\theta$ some non-negative and continuous function in $\mathbb{R}$. We find minimal additional requirements to $\theta$ under which for a given $p \in [1,\infty]$ there exists an absolute positive constant $c(p)$ such that the following inequality holds: \begin{equation*} \sup_{x \in \mathbb{R}} \theta(x)|y(x)| \leq c(p) \|f\|_{L_{p}(\mathbb{R})} \qquad \forall f \in L_{p}(\mathbb{R}). \end{equation*}