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

Sándor Kajántó, Alexandru Kristály, Ioan Radu Peter, Wei Zhao

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.

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

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.
Paper Structure (12 sections, 27 theorems, 144 equations)

This paper contains 12 sections, 27 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of main results
  4. Applications I: Additive Hardy-type inequalities via Riccati pairs
  5. Caccioppoli inequalities
  6. Hardy inequalities and their improvements on Cartan-Hadamard manifolds
  7. Spectral estimates on Riemannian manifolds
  8. Interpolation: Hardy inequality versus McKean spectral gap
  9. Ghoussoub-Moradifam-type inequalities
  10. Applications II: Multiplicative Hardy-type inequalities
  11. Sharp uncertainty principles
  12. Caffarelli-Kohn-Nirenberg inequalities

Key Result

Theorem 1.1

Let $(M,g)$ be a complete, non-compact $n$-dimensional Riemannian manifold, with $n\geq 2$. Let $\Omega\subseteq M$ be a domain, $p>1$, and $\rho\in W_{\rm loc}^{1,p}(\Omega)$ be a positive function with $|\nabla_g\rho|=1$$\,\mathrm{d} v_g$-a.e. in $\Omega$. Let $G\colon(0,\sup_\Omega\rho)\to \mathb and $H(0)=H'(0)=0$. The following inequalities hold.

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 57 more