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Hardy, Sobolev, and CLR Inequalities

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)

Abstract

The Hardy and Sobolev inequalities are of fundamental importance in many branches of mathematical analysis and mathematical physics, and have been intensively studied since their discovery. A rich theory has been developed with the original inequalities on \((0,\infty )\) extended and refined in many ways, and an extensive literature on them now exists. We shall be focusing throughout the book on versions of the inequalities in L p spaces, with \(1 < p < \infty \). In this chapter we shall be mainly concerned with the inequalities in \((0,\infty )\) or \(\mathbb{R}^{n},\ n \geq 1\). Later in the chapter we shall also discuss the CLR (Cwikel, Lieb, Rosenbljum) inequality, which gives an upper bound to the number of negative eigenvalues of a lower semi-bounded Schrödinger operator in \(L^{2}(\mathbb{R}^{n})\). This has a natural place with the Hardy and Sobolev inequalities as the three inequalities are intimately related, as we shall show. Where proofs are omitted, e.g., of the Sobolev inequality, precise references are given, but in all cases we have striven to include enough background analysis to enable a reader to understand and appreciate the result.

Keywords

Sobolev Inequality Extension Property Hardy Inequality Markov Generator Truncation Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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