2019/07/30 by Xavier Cabré, Xavier Cabre, Fernando Charro +2
Mathematics · #42B37 #46E35 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #math.AP #math.CA #msc:42B37 #msc:46E35
paper · pdf · doi:10.48550/arxiv.1907.12982
30 pages, 4 figures. Version 2 contains new references and some comments to them. A few minor misprints have been corrected in version 3
openalex publication_date 2019/07/30 · arxiv created 2020/12/22 · arxiv updated 2020/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the following natural question that, apparently, has not been well addressed in the literature: Given functions u with support in the unit ball B1⊂ℝn and with gradient in the Morrey space Mp,λ(B1), where 1<p<λ<n, what is the largest range of exponents q for which necessarily u∈ Lq(B1)? While David R. Adams proved in 1975 that this embedding holds for q≤λp/(λ-p), an article from 2011 claimed the embedding in the larger range q<n p/(λ-p). Here we disprove this last statement by constructing a function that provides a counterexample for q>λp/(λ-p). The function is basically a negative power of the distance to a set of Hausdorff dimension n-λ. When λ∉ℤ, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent q can go up to np/(λ-p).