Kajanto, S., Kristaly, A., Peter, I.R. & Zhao, W. (In press) Mathematische Annalen [Matematică, Q1]

Publicat: 12 Aprilie 2024

Kajanto, S., Kristaly, A., Peter, I.R. & Zhao, W. (In press) A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities. Mathematische Annalen.

DOI: https://doi.org/10.1007/s00208-024-02827-7

✓ Publisher: Springer
✓ Categories: Mathematics
✓ Article Influence Score (AIS): 1.671 (2022) / Q1

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. Ann., 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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