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A Short Note on Universal Inequalities for Certain Potential Operators

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Abstract

We give universal inequalities for the norm of certain potential operators in terms of the least non-zero Neumann eigenvalue.

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References

  1. Anderson, J.M., Khavinson, D., Lomonosov, V.: Spectral properties of some integral operators arising in potential theory. Q. J. Math. Oxf. Ser. (2) 43(172), 387–407 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ashbaugh, M.S., Benguria, R.D.: Universal bounds for the low eigenvalues of Neumann Laplacians in n dimensions. SIAM J. Math. Anal. 24(3), 557–570 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benguria, R.M., Helmut, L., Loewe, B.: Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator. Bull. Math. Sci. 2(1), 1–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Friedlander, L.: Some inequalities between Dirichlet and Neumann eigenvalues. Arch. Ration. Mech. Anal. 116, 153–160 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Frank, R.L., Laptev, A.: Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group. Int. Math. Res. Not. IMRN 15, 2889–2902 (2010)

  6. Levine, H.A., Weinberger, H.F.: Inequalities between Dirichlet and Neumann eigenvalues. Arch. Ration. Mech. Anal. 94(3), 193–208 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Payne, L.E.: Inequalities for eigenvalues of membranes and plates. J. Ration. Mech. Anal. 4, 517–529 (1955)

    MathSciNet  MATH  Google Scholar 

  8. Pólya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. (3) 11, 419–433 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rozenblum, G., Ruzhansky, M., Suragan, D.: Isoperimetric inequalities for Schatten norms of Riesz potentials. J. Funct. Anal. 271(1), 224–239 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ruzhansky, M., Suragan, D.: Isoperimetric inequalities for the logarithmic potential operator. J. Math. Anal. Appl. 434(2), 1676–1689 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3, 343–356 (1954)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

I am indebted to Prof. Durvudkhan Suragan for numerous discussions during preparation of this work. I also would like to thank Prof. Christian Remling for pointing out the example mentioned below Corollary 2.4.

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Correspondence to Seyed M. Zoalroshd.

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Communicated by Dr. Colombo and Dr. Naboko.

This paper is dedicated to the memory of my mother.

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Zoalroshd, S.M. A Short Note on Universal Inequalities for Certain Potential Operators. Complex Anal. Oper. Theory 11, 1463–1466 (2017). https://doi.org/10.1007/s11785-017-0694-0

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