A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities
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| Format: | Preprint |
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2023
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| _version_ | 1866914679344857088 |
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| author | Kajántó, Sándor Kristály, Alexandru Peter, Ioan Radu Zhao, Wei |
| author_facet | Kajántó, Sándor Kristály, Alexandru Peter, Ioan Radu Zhao, Wei |
| contents | We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math.A nn., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_09965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities Kajántó, Sándor Kristály, Alexandru Peter, Ioan Radu Zhao, Wei Analysis of PDEs Differential Geometry Functional Analysis We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math.A nn., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed. |
| title | A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities |
| topic | Analysis of PDEs Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2303.09965 |