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arXiv · 2303.09965

A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities

Sándor Kajántó·Alexandru Kristály·Ioan Radu Peter·Wei Zhao
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Abstract

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.

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Sándor Kajántó, Alexandru Kristály, Ioan Radu Peter, Wei Zhao. 2023-03-17. A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities. https://arxiv.org/abs/2303.09965

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