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Some Results Concerning Hardy and Hardy Type Inequalities

In: Handbook of Functional Equations

Author

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  • Nikolaos B. Zographopoulos

    (Hellenic Army Academy)

Abstract

We review some recent results concerning functional aspects of the Hardy and Hardy type inequalities. Our main focus is the formulation of such inequalities, for functions having bad behavior at the singularity points. It turns out that Hardy’s singularity terms appear in certain cases as a loss to the Hardy’s functional, while in other cases are additive to it. Surprisingly, in the latter case, Hardy’s functional may be negative. Thus, the validity of the Hardy’s inequality is actually based on these singularity terms. We also discuss the two topics: nonexistence of $H_0^1$ minimizers and improved Hardy–Sobolev inequalities. These topics may be seen as a consequence of the connection of the Hardy and Hardy type inequalities with the Sobolev inequality defined in the whole space.

Suggested Citation

  • Nikolaos B. Zographopoulos, 2014. "Some Results Concerning Hardy and Hardy Type Inequalities," Springer Optimization and Its Applications, in: Themistocles M. Rassias (ed.), Handbook of Functional Equations, edition 127, pages 535-555, Springer.
    • Handle: RePEc:spr:spochp:978-1-4939-1246-9_20
    DOI: 10.1007/978-1-4939-1246-9_20
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