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

TL;DR

This paper develops a new method combining the Caffarelli-Silvestre extension and Duhamel formulas to derive exact identities for fractional Laplacian operators on Riemannian manifolds, leading to novel formulas and inequalities.

Contribution

It introduces a new approach for fractional Laplacian analysis, producing the first explicit fractional Bochner's formula and exact remainders in key inequalities.

Findings

Derived a pointwise fractional Leibniz rule.

Established a fractional Bochner's formula with Ricci curvature.

Provided exact remainders in Córdova-Córdova and Kato inequalities.

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 C\'ordoba-C\'ordoba and Kato inequalities for the fractional Laplacian. All these formulas are new even in the Euclidean space.