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

• Main Authors: Kajántó, Sándor, Kristály, Alexandru, Peter, Ioan Radu, Zhao, Wei
• Format: Journal Article
• Language: English
• Published: 17-03-2023
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Differential Geometry; Mathematics - Functional Analysis
• DOI: 10.48550/arxiv.2303.09965

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.