2018/07/12 by Jonas Azzam, Azzam, Jonas
Mathematics · #26D15 #28A75 #46E35 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1807.04534
openalex publication_date 2018/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that if 0<t<s\≤ n-1, \Ω\⊆ \ℝn with lower\ns-content regular complement, and z\∈ \Ω, there is a chord-arc domain\n\Ωz\⊆ \Ω with center z so that\n mathscrHt\∞(\∂\Ωz\∩ \∂\Ω) gtrsimt\n textrmdist(z,\Ωc)t. This was originally shown by Koskela, Nandi,\nand Nicolau with John domains in place of chord-arc domains when n=2, s=1,\nand \Ω is a simply connected planar domain.\n Domains satisfying the conclusion of this result support (p,\β)-Hardy\ninequalities for \β<p-n+t by a result of Koskela and Lehrb "ack;\nLehrb "ack also showed that s-content regularity of the complement for some\ns>n-p+\β was necessary. Thus, the combination of these results gives a\ncharacterization of which domains support pointwise (p,\β)-Hardy\ninequalities for \β<p-1 in terms of lower content regularity.\n