2008/01/11 by V. Eiderman, Vladimir Eiderman, F. Nazarov +6
Mathematics · #28A78 #30C85 #31B15 #31C45 (Secondary) #42B20 (Primary) #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #math.AP #msc:28A78 #msc:30C85 #msc:31B15 #msc:31C45 #msc:42B20
paper · pdf · doi:10.48550/arxiv.0801.1855
33 pages
openalex publication_date 2008/01/11 · arxiv created 2008/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are many interesting problems about the electrostatic potential of finitely many charges. We consider one of them concerning the intensity of the field, in other words, about the magnitude of the gradient of this potential. We want to give a sharp estimate of the size of the set of points where this gradient is large. Of course we want the estimate to be sharp in number N of charges. The size will be measured by the Hausdorff content with various gauge functions. Such a setting allows us to consider a wide class of measures (not necessarily with finitely many charges). The main technique will be Calderón-Zygmund capacities and nonhomogeneous Calderón-Zygmund operators. Here we establish a relationship between various types of capacities with singular kernels (e. g. analytic capacity, lipschitz harmonic capacity, etc) and non-linear capacity from the theory of potential á la Adams, Hedberg, Havin, Maz'ya, Wolff. "Capacitary" part of the paper extends the theorem of Mateu, Prat and Verdera [J. reine und angew. Math., 578 (2005), 201--223]. "Size estimates" part of the paper extends the theorem of M. Anderson and V. Eiderman [Annals of Math., 163 (2005), 1057--1076]. The difficulty lies in the fact that we cannot use Menger's curvature anymore because we are working in spaces of dimension bigger than two.