Embeddings between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces

2016/06/21 by Амиран Гогатишвили, Gogatishvili, Amiran, Rza Mustafayev +3
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Primary 46E30 #Secondary 26D10

paper · pdf · doi:10.48550/arxiv.1606.06745

openalex publication_date 2016/06/21 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

Abstract

In this paper embeddings between weighted complementary local Morrey-type spaces ^^{\bf c} LM_pθ,ω(\mathbb Rⁿ,v) and weighted local Morrey-type spaces LM_pθ,ω(\mathbb Rⁿ,v) are characterized. In particular, two-sided estimates of the optimal constant c in the inequality ( ∫₀^∞ ( ∫_B(0,t) f(x)^p₂v₂(x) dx )^{(q₂)/(p₂)} u₂(t) dt)^(1)/(q₂) ≤ c ( ∫₀^∞ ( ∫_ ^^{\bf c} B(0,t) f(x)^p₁ v₁(x) dx)^{(q₁)/(p₁)} u₁(t) dt)^(1)/(q₁) are obtained, where p₁, p₂, q₁, q₂ ∈ (0,∞), p₂ ≤ q₂ and u₁, u₂ and v₁, v₂ are weights on (0,∞) and \mathbb Rⁿ, respectively. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted local Morrey-type and complementary local Morrey-type spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of the iterated Hardy-type inequalities.

Citations

• On the solutions of quasi-linear elliptic partial differential equations

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