Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem

N. A. Chernyavskaya; L. A. Shuste

Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem

N. A. Chernyavskaya, L. A. Shuste

Research output: Contribution to journal › Article › peer-review

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Abstract

We consider the equation -y′(x) + q(x)y(x) = f(x); x εℝ (0.1) where f ε L_p(ℝ) p ε [1,∞ ] (L _∞ (ℝ) := C(ℝ) and 0≤ q ε L₁ ^loc (ℝ): We assume that equation (0.1) is correctly solvable in L_p(ℝ). Let y εL_p(ℝ) be a solution of (0.1). In the present paper we find minimal requirements for the weight function r ε L₁^loc p (ℝ) under which the following estimate holds: ∥ry∥_p ≤ c(p)∥f∥_p, ∀f ε L_p (ℝ) with an absolute constant c(p) ε (0, ∞).

Original language	English
Pages (from-to)	467-504
Number of pages	38
Journal	Differential and Integral Equations
Volume	25
Issue number	5-6
State	Published - May 2012
Externally published	Yes

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cited By 2

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title = "Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem",

abstract = "We consider the equation -y′(x) + q(x)y(x) = f(x); x εℝ (0.1) where f ε Lp(ℝ) p ε [1,∞ ] (L ∞ (ℝ) := C(ℝ) and 0≤ q ε L1 loc (ℝ): We assume that equation (0.1) is correctly solvable in Lp(ℝ). Let y εLp(ℝ) be a solution of (0.1). In the present paper we find minimal requirements for the weight function r ε L1loc p (ℝ) under which the following estimate holds: ∥ry∥p ≤ c(p)∥f∥p, ∀f ε Lp (ℝ) with an absolute constant c(p) ε (0, ∞).",

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AU - Shuste, L. A.

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PY - 2012/5

Y1 - 2012/5

N2 - We consider the equation -y′(x) + q(x)y(x) = f(x); x εℝ (0.1) where f ε Lp(ℝ) p ε [1,∞ ] (L ∞ (ℝ) := C(ℝ) and 0≤ q ε L1 loc (ℝ): We assume that equation (0.1) is correctly solvable in Lp(ℝ). Let y εLp(ℝ) be a solution of (0.1). In the present paper we find minimal requirements for the weight function r ε L1loc p (ℝ) under which the following estimate holds: ∥ry∥p ≤ c(p)∥f∥p, ∀f ε Lp (ℝ) with an absolute constant c(p) ε (0, ∞).

AB - We consider the equation -y′(x) + q(x)y(x) = f(x); x εℝ (0.1) where f ε Lp(ℝ) p ε [1,∞ ] (L ∞ (ℝ) := C(ℝ) and 0≤ q ε L1 loc (ℝ): We assume that equation (0.1) is correctly solvable in Lp(ℝ). Let y εLp(ℝ) be a solution of (0.1). In the present paper we find minimal requirements for the weight function r ε L1loc p (ℝ) under which the following estimate holds: ∥ry∥p ≤ c(p)∥f∥p, ∀f ε Lp (ℝ) with an absolute constant c(p) ε (0, ∞).

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SN - 0893-4983

VL - 25

SP - 467

EP - 504

JO - Differential and Integral Equations

JF - Differential and Integral Equations

IS - 5-6

ER -

Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem

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