Archiv der Mathematik

, Volume 112, Issue 6, pp 645–648 | Cite as

Upper and lower bounds for the Hilbert–Schmidt norm of a potential operator

  • Seyed ZoalroshdEmail author


We give upper and lower bounds for the Hilbert–Schmidt norm of logarithmic potential on a planar domain in terms of its area and inradius.


Hilbert–Schmidt norm Logarithmic potential Polygon 

Mathematics Subject Classification

Primary 35S15 Secondary 35P99 


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I am very grateful to the referee for his/her various useful suggestions specially for introducing to the works of Exner, Harrell, Krejčiřík, Loss and Lotoreichik related to the topic of this note.


  1. 1.
    Arazy, J., Khavinson, D.: Spectral estimates of Cauchy’s transform in \(L^2(\Omega )\). Integral Equ. Oper. Theory 15(6), 901–919 (1992)CrossRefzbMATHGoogle Scholar
  2. 2.
    Exner, P., Harrell, E.M., Loss, M.: Inequalities for means of chords, with application to isoperimetric problems. Lett. Math. Phys. 75(3), 225–233 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Exner, P., Lotoreichik, V.: A spectral isoperimetric inequality for cones. Lett. Math. Phys. 107(4), 717–732 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Freitas, P., Krejčiřík, D.: A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains. Proc. Am. Math. Soc. 136(8), 2997–3006 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Krejčiřík, D., Lotoreichik, V.: Optimisation of the lowest Robin eigenvalue in the exterior of a compact set. J. Convex Anal. 25, 319–337 (2018)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Lotoreichik, V.: Spectral isoperimetric inequalities for singular interactions on open arcs. Applicable Analysis, 1–10 (2018)Google Scholar
  7. 7.
    Ruzhansky, M., Suragan, D.: Isoperimetric inequalities for the logarithmic potential operator. J. Math. Anal. Appl. 434(2), 1676–1689 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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