2016/07/16 by N. A. Chernyavskaya, Chernyavskaya, N. A., L. A. Shuster +1
Mathematics · #34B05 #34B24 #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1607.04797
openalex publication_date 2016/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the equation -y''(x)+q(x)y(x)=f(x), x∈ \mathbb R where f ∈ Lploc(\mathbb R), p ∈ [1,∞) and 0 < q ∈ L1loc(\mathbb R). By a solution of this equation we mean any function y, absolutely continuous together with its derivative and satisfying the equation almost everywhere in \mathbb R. Let positive and continuous functions μ(x) and θ(x) for x ∈ \mathbb R be given. Let us introduce the spaces Lp(\mathbb R,μ) = \f ∈ Lploc(\mathbb R): ||f||Lp(\mathbb R,μ)p =∫-∞^∞|μ(x)f(x)|p dx < ∞\, Lp(\mathbb R,θ) = \f∈ Lploc(\mathbb R):||f||Lp(\mathbb R,θ)p = ∫-∞^∞|θ(x)f(x)|p dx < ∞\. In the present paper, we obtain requirements to the functions μ,θ and q under which 1) for every function f ∈ Lp(\mathbb R,θ) there exists a unique solution of the equation y ∈ Lp(\mathbb R,μ) ; 2) there is an absolute constant c(p) ∈ (0,∞) such that regardless of he choice of a function f ∈ Lp(\mathbb R,θ) the solution of the equation satisfies the inequality ‖y‖Lp(\mathbb R,μ) lt; c(p)‖f‖Lp(\mathbb R,θ).