Variable exponent Hardy-type inequalities in ℝⁿ

2015/03/25 by Sylwia Dudek, Dudek, Sylwia, Iwona Skrzypczak +1
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1503.07317

openalex publication_date 2015/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate further the weighted p(x)-Hardy inequality with the additional term of the form ∫_Ω|ξ|^p(x)μ_1,β (dx) \leqslant ∫_Ω|∇ ξ|^p(x)μ_2,β (dx)+∫_Ω|ξlog ξ |^p(x) μ_3,β (dx), holding for Lipschitz functions compactly supported in Ω⊆ℝⁿ. The involved measures depend on a certain solution to the partial differential inequality involving p(x)-Laplacian -Δ_p(x) u\geqslant Φ, where Φ is a given locally integrable function, and u is defined on an open and not necessarily bounded subset Ω⊆ℝⁿ , and a certain parameter β. We focus on the n-dimensional case giving some examples. Moreover, we compare our inequalities with the existing in the literature.

Related

• Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem
• On necessary and sufficient conditions for the variable exponent Hardy type inequality
• A note on Hardy-type inequalities in variable exponent Lebesgue spaces
• A new approach to norm inequalities on weighted and variable Hardy\n spaces
• On a weighted variable spaces L_{p(x), ω} for 0< p(x)< 1 and weighted Hardy inequality