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

N. A. Chernyavskaya; L. A. Shuster

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 2, page 423-448
  • ISSN: 0392-4041

Abstract

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We consider the equation - y ′′ ⁢ ( x ) + q ⁢ ( x ) ⁢ y ⁢ ( x ) = f ⁢ ( x ) , x ∈ ℝ , where f ∈ L p ⁢ ( ℝ ) , p ∈ [ 1 , ∞ ] ( L ∞ ⁢ ( ℝ ) := C ⁢ ( ℝ ) ) and 0 ≤ q ∈ L 1 loc ( ℝ ) ; ∃ a > 0 : inf x ∈ ℝ ∫ x - a x + a q ( t ) d t > 0 , (Condition (2) guarantees correct solvability of (1) in class L p ⁢ ( ℝ ) , p ∈ [ 1 , ∞ ] .) Let y be a solution of (1) in class L p ⁢ ( ℝ ) , p ∈ [ 1 , ∞ ] , and θ some non-negative and continuous function in ℝ . We find minimal additional requirements to θ under which for a given p ∈ [ 1 , ∞ ] there exists an absolute positive constant c ⁢ ( p ) such that the following inequality holds: sup x ∈ ℝ ⁡ θ ⁢ ( x ) ⁢ | y ⁢ ( x ) | ≤ c ⁢ ( p ) ⁢ ∥ f ∥ L p ⁢ ( ℝ )    ∀ f ∈ L p ⁢ ( ℝ ) .

How to cite

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Chernyavskaya, N. A., and Shuster, L. A.. "Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation." Bollettino dell'Unione Matematica Italiana 5.2 (2012): 423-448. <http://eudml.org/doc/290967>.

@article{Chernyavskaya2012,
abstract = {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*\}},
author = {Chernyavskaya, N. A., Shuster, L. A.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {423-448},
publisher = {Unione Matematica Italiana},
title = {Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation},
url = {http://eudml.org/doc/290967},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Chernyavskaya, N. A.
AU - Shuster, L. A.
TI - Integral Inequalities for the Principal Fundamental System of Solutions of a Homogeneous Sturm-Liouville Equation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/6//
PB - Unione Matematica Italiana
VL - 5
IS - 2
SP - 423
EP - 448
AB - 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*}
LA - eng
UR - http://eudml.org/doc/290967
ER -

References

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