Advertisement

Mathematical Notes

, Volume 86, Issue 5–6, pp 753–766 | Cite as

On the possibility of strengthening the Lieb-Thirring inequality

  • D. S. Barsegyan
Article
  • 67 Downloads

Abstract

In 1976, Lieb and Thirring obtained an upper bound for the square of the normon L 2(ℝ2) of the sum of the squares of functions from finite orthonormal systems via the sum of the squares of the norms of their gradients. Later, a series of Lieb-Thirring inequalities for orthonormal systems was established by many authors. In the present paper, using the standard theory of functions, we prove Lieb-Thirring inequalities, which have applications in the theory of partial differential equations.

Key words

Lieb-Thirring inequality orthonormal system function theory Fourier transform partial differential equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. H. Lieb and W. Thirring, “Inequalities for the moments of the eigenvalues the Schrödinger Hamiltonian and their relation to Sobolev inequalities,” in Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann (Princeton Univ. Press, Princeton, NJ, 1976), pp. 269–303.Google Scholar
  2. 2.
    B. S. Kashin, “On a class of inequalities for orthonormal systems,” Mat. Zametki 80(2), 204–208 (2006) [Math. Notes 80 (1–2), 199–203 (2006)].MathSciNetGoogle Scholar
  3. 3.
    D. S. Barsegyan, “On inequalities of Lieb-Thirring type,” Mat. Zametki 82(4), 504–514 (2007) [Math. Notes 82 (3–4), 451–460 (2007)].MathSciNetGoogle Scholar
  4. 4.
    S. V. Astashkin, “The Lieb-Thirring inequality for L p-norms,” Mat. Zametki 83(2), 163–169 (2008) [Math. Notes 83 (1–2), 145–151 (2008)].MathSciNetGoogle Scholar
  5. 5.
    B. S. Kashin and A. A. Saakyan, Orthogonal Series (Izd. AFTs, Moscow, 1999) [in Russian].Google Scholar
  6. 6.
    S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems (Nauka, Moscow, 1977) [in Russian].Google Scholar
  7. 7.
    I. P. Natanson, Theory of Functions of a Real Variable (Nauka, Moscow, 1974) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • D. S. Barsegyan
    • 1
  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations