2019/05/16 by T. V. Anoop, Anoop, T. V., Ujjal Das +1 · 1 citation
Mathematics · #28A12 #28A33 #35A23 #35J20 #46E30 #46E35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Banach space #Combinatorics #Compact space #Discrete mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical analysis #Mathematics #Nabla symbol #Nonlinear Partial Differential Equations #Norm (philosophy) #Omega #Physics #Quantum mechanics #Sobolev space #math.AP #math.FA #msc:28A12 #msc:28A33 #msc:35A23 #msc:35J20 #msc:46E30 #msc:46E35
paper · pdf · doi:10.48550/arxiv.1905.06921
27 pages, Changes in the hypothesis of Theorem 1.4 and Theorem 1.5
openalex publication_date 2019/05/16 · arxiv created 2021/02/10 · arxiv updated 2021/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For p ∈ (1,N) and Ω⊆ ℝN open, the Beppo-Levi space D1,p0(Ω) is the completion of Cc∞(Ω) with respect to the norm ( ∫Ω|∇ u|p )^ (1)/(p). Using the p-capacity, we define a norm and then identify the Banach function space H(Ω) with the set of all g in L1loc(Ω) that admits the following Hardy-Sobolev type inequality: ∫Ω |g| |u|p ≤ C ∫Ω |∇ u|p, ∀ u ∈ D1,p0(Ω), for some C>0. Further, we characterize the set of all g in H(Ω) for which the map G(u)= ∫Ω g |u|p is compact on D1,p0(Ω). We use a variation of the concentration compactness lemma to give a sufficient condition on g∈ H(Ω) so that the best constant in the above inequality is attained in D1,p0(Ω).