English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present a generic functional inequality on Riemannian manifolds, both inadditive and multiplicative forms, that produces well known and genuinely newHardy-type inequalities. For the additive version, we introduce Riccati pairsthat extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl.Acad. Sci. USA, 2008 & Math.A nn., 2011). This concept enables us to give veryshort/elegant proofs of a number of celebrated functional inequalities onRiemannian manifolds with sectional curvature bounded from above by simplysolving a Riccati-type ODE. Among others, we provide alternative proofs forCaccioppoli inequalities, Hardy-type inequalities and their improvements,spectral gap estimates, interpolation inequalities, andGhoussoub-Moradifam-type weighted inequalities. Concerning the multiplicativeform, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e.,Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principlesand Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidityphenomena are also discussed.

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