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arXiv · 2510.01942

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

The Anh Bui
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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.

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The Anh Bui. 2025-10-02. Bilinear and Fractional Leibniz Rules Beyond Euclidean Spaces: Weighted Besov and Triebel--Lizorkin Estimates. https://arxiv.org/abs/2510.01942

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