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

Chernyavskaya, N. A.; Shuster, L. A.

Bollettino dell'Unione Matematica Italiana, Tome 5 (2012), p. 423-448 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé

We consider the equation $-y^{\prime\prime}(x)+q(x)y(x)=f(x),\qquad x\in\mathbb{R},$ where $f\in L_{p}(\mathbb{R})$ , $p\in[1,\infty]$ ( $L_{\infty}(\mathbb{R}):=C(\mathbb{R})$ ) and $0\leq q\in L_{1}^{\text{loc}}(\mathbb{R});\qquad\exists a>0:\inf_{x\in\mathbb{% R}}\int_{x-a}^{x+a}q(t)\,dt>0,$ (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: $\sup_{x\in\mathbb{R}}\theta(x)|y(x)|\leq c(p)\|f\|_{L_{p}(\mathbb{R})}\qquad% \forall f\in L_{p}(\mathbb{R}).$

Publié le : 2012-06-01

MR 2977257 | Zbl 1260.34063

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     author = {N. A. Chernyavskaya and L. A. Shuster},
     title = {Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5},
     year = {2012},
     pages = {423-448},
     zbl = {1260.34063},
     mrnumber = {2977257},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2012_9_5_2_423_0}
}

Chernyavskaya, N. A.; Shuster, L. A. Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation. Bollettino dell'Unione Matematica Italiana, Tome 5 (2012) pp. 423-448. http://gdmltest.u-ga.fr/item/BUMI_2012_9_5_2_423_0/

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