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A new Duhamel-type principle with applications to geometric (in)equalities

Michele Caselli, Luca Gennaioli

Abstract

We introduce a simple new method, based on the Caffarelli-Silvestre extension and a Duhamel-type formula, to derive exact pointwise identities for fractional commutators and nonlinear compositions associated with the fractional Laplacian on general Riemannian manifolds. As applications, we obtain a pointwise fractional Leibniz rule, a fractional Bochner's formula with an explicit Ricci curvature term, apparently the first of this kind, and exact remainders in the Córdoba-Córdoba and Kato inequalities for the fractional Laplacian. All these formulas are new even in the Euclidean space.

A new Duhamel-type principle with applications to geometric (in)equalities

Abstract

We introduce a simple new method, based on the Caffarelli-Silvestre extension and a Duhamel-type formula, to derive exact pointwise identities for fractional commutators and nonlinear compositions associated with the fractional Laplacian on general Riemannian manifolds. As applications, we obtain a pointwise fractional Leibniz rule, a fractional Bochner's formula with an explicit Ricci curvature term, apparently the first of this kind, and exact remainders in the Córdoba-Córdoba and Kato inequalities for the fractional Laplacian. All these formulas are new even in the Euclidean space.

Paper Structure

This paper contains 10 sections, 13 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Relation to prior work
  3. Main results
  4. Applications
  5. Extension framework and the Duhamel-type formula
  6. Proof of the Duhamel-type formula
  7. Proof of the main results
  8. Pointwise fractional Leibniz rule: proof of Theorem \ref{['thm: baby Boch Gamma_1']}
  9. Fractional Bochner's formula: proof of Theorem \ref{['thm: Bochner ricci']}
  10. Remainder in the Córdoba-Córdoba inequality: proof of Theorem \ref{['thm: C-C general phi']}

Key Result

Proposition 1.1

Let $s\in (0,1)$, and $F \in L^1(\widetilde{M})$, $V \in \widehat{H}^1_w (\widetilde{M} )$ be smooth functions, with $V$ solution of Then Here $\mathcal{P}^{(s)}_z$ denotes the fractional Poisson operator eq: frac Poisson def, which acts on $F(\cdot, z)$ for fixed $z>0$.

Theorems & Definitions (26)

  • Proposition 1.1: Duhamel-type formula for the extension
  • Theorem 1.2: Pointwise fractional Leibniz rule
  • Theorem 1.3: Fractional Bochner's formula
  • Corollary 1.4
  • Theorem 1.5: Exact remainder in the Córdoba-Córdoba inequality
  • Proposition 1.6
  • Corollary 1.7: Pointwise Stroock-Varopoulos identity
  • Theorem 1.8: Exact remainder in the Kato inequality
  • Remark 2.1
  • proof : Proof of Proposition \ref{['prop: rep formula solutions']}
  • ...and 16 more