The Operator Norm of Paraproducts on Hardy Spaces

Shahaboddin Shaabani

Published 2024-02-20, updated 2024-08-09Version 2

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 \(\Pi_g\), of the form \[ \Pi_{g}(f) := \sum_{j \in \mathbb{Z}} (\varphi_{2^{-j}} * f) \cdot \Delta_jg, \] 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.