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Hardy type inequalities in [Formula: see text] with sharp remainders
Norisuke Ioku1, Michinori Ishiwata2, Tohru Ozawa3
1Graduate School of Science and Engineering, Ehime University, Ehime, 790-8577 Japan.
Journal of Inequalities and Applications
|January 24, 2017
Summary
This study provides explicit sharp remainder terms for Hardy inequalities in [Formula: see text] and [Formula: see text]. These findings clarify Hardy type inequalities and the nonexistence of nontrivial extremals.
Area of Science:
- Mathematical Analysis
- Functional Analysis
- Inequalities
Background:
- Hardy inequalities are fundamental in analysis.
- Understanding the precise error terms (remainders) is crucial for deeper insights.
- Previous work has established various forms of Hardy inequalities.
Purpose of the Study:
- To explicitly derive sharp remainder terms for standard Hardy inequalities.
- To provide an exact understanding of Hardy type inequalities through these remainder terms.
- To investigate the conditions for the nonexistence of nontrivial extremals.
Main Methods:
- Direct derivation of remainder terms.
- Analysis within the framework of equalities.
- Exploitation of the properties of remainder terms to infer extremal behavior.
Main Results:
- Explicit formulas for sharp remainder terms in Hardy inequalities are presented.
- The remainder terms offer a precise characterization of the inequalities.
- Conditions are established demonstrating the nonexistence of nontrivial extremals.
Conclusions:
- The derived remainder terms provide a complete understanding of Hardy inequalities.
- The results offer exact conditions for the nonexistence of nontrivial extremals.
- This work advances the study of integral inequalities and their extremal properties.
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