Bilinear and Fractional Leibniz Rules Beyond Euclidean Spaces: Weighted Besov and Triebel--Lizorkin Estimates

TL;DR

The paper tackles fractional Leibniz rules for products under fractional differentiation in spaces of homogeneous type, extending Kato–Ponce-type estimates to settings where Fourier analysis is unavailable. It introduces a unified bilinear spectral multiplier framework tied to a nonnegative self-adjoint operator $L$ and weighted Besov/Triebel–Lizorkin spaces, proving sharp bilinear bounds that hold under Gaussian heat-kernel bounds, Hölder continuity, and a higher-order derivative condition (A3). The main theorem unifies Euclidean and non-Euclidean cases, recovering classical results for $L=-\,\Delta$ while simultaneously extending to nilpotent Lie groups, Grushin operators, and Hermite operators with weights in $A_\infty$, thereby broadening applicability to nonlinear PDE analysis on spaces of homogeneous type. The work further demonstrates applications to scattering in PDEs by linking long-time limits to operator-valued bilinear forms, yielding new weighted estimates crucial for nonlinear dynamics in non-Euclidean geometries.

Abstract

We establish fractional Leibniz rules in weighted settings for nonnegative self-adjoint operators on spaces of homogeneous type. Using a unified method that avoids Fourier transforms, we prove bilinear estimates for spectral multiplier on weighted Hardy, Besov and Triebel-Lizorkin spaces. Our approach is flexible and applies beyond the Euclidean setting-covering, for instance, nilpotent Lie groups, Grushin operators, and Hermite expansions-thus extending classical Kato-Ponce inequalities. The framework also yields new weighted bilinear estimates including fractional Leibniz rules for Hermite, Laguerre, and Bessel operator, with applications to scattering formulas and related PDE models.

Theorems & Definitions (64)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Proposition 2.7