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Mathematische Annalen

, Volume 373, Issue 1–2, pp 253–285 | Cite as

Failure of \(L^2\) boundedness of gradients of single layer potentials for measures with zero low density

  • José M. Conde-Alonso
  • Mihalis MourgoglouEmail author
  • Xavier Tolsa
Article
  • 53 Downloads

Abstract

Consider a totally irregular measure \(\mu \) in \({{\mathbb {R}}}^{n+1}\), that is, the upper density \(\limsup _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) is positive \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\), and the lower density \(\liminf _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) vanishes \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\). We show that if \(T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu (y)\) is an operator whose kernel \(K(\cdot ,\cdot )\) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then \(T_\mu \) is not bounded in \(L^2(\mu )\). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Departamento de MatemàticasUniversidad del País VascoLeioaSpain
  3. 3.Ikerbasque, Basque Foundation for ScienceBilbaoSpain
  4. 4.ICREABarcelonaCatalonia
  5. 5.Departament de Matemàtiques and BGSMathUniversitat Autònoma de BarcelonaBellaterra, BarcelonaCatalonia

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