The Operator Norm of Paraproducts on Hardy Spaces

TL;DR

This work characterizes the operator norm of paraproducts on Hardy spaces, establishing that the norm of a Fourier paraproduct $Π_g$ acting from $H^p( d^n)$ to $ h^q( d^n)$ is comparable to the homogeneous Sobolev/Lipschitz norm of the symbol $g$, specifically $ig r{Π_g}ig_{H^p o h^q}\, hicksim \,ig r{g}_{ hdot^r}$ with $rac{1}{q}=rac{1}{p}+rac{1}{r}$. The paper also proves dyadic analogues for dyadic paraproducts on dyadic Hardy spaces, and extends the Fourier-paraproduct results to a broad range of target spaces via a sublinear square-function control $ S_{g,}$, linking the operator norm to $ hdot^eta$-type dual spaces and $BMO$. The key method is a unifying sparse domination and testing approach, combining dyadic reductions, maximal function estimates, and duality arguments to obtain sharp, uniform bounds and to show optimality of the exponents. Overall, the results yield sharp, non-improvable estimates for paraproduct operators across both dyadic and continuous settings, with precise dependence on the symbol norms and clear pathways for multi-parameter extensions. The findings have implications for nonlinear PDE analysis and the study of para-differential calculi under minimal regularity assumptions.

Abstract

For a tempered distribution $g$, and $0 < p, q, r < \infty$ with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$, we show that the operator norm of a Fourier paraproduct $Π_g$, of the form \[ Π_{g}(f) := \sum_{j \in \mathbb{Z}} (\varphi_{2^{-j}} * f) \cdot Δ_jg, \] from $H^p(\mathbb{R}^n)$ to $\dot{H}^q(\mathbb{R}^n)$ is comparable to $\|g\|_{\dot{H}^r(\mathbb{R}^n)}$. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces.

Theorems & Definitions (36)

• Definition 2.1
• Lemma 2.2
• Theorem : Lerner, Lorist, Ombrosi
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Theorem : A
• Proposition 2.8