The Operator Norm of Paraproducts on Hardy Spaces

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 ${\Pi }_g$, of the form \[ Π_{g}(f) := \sum_{j \in \mathbb{Z}} (φ_{2^{-j}} * f) \cdot Δ_jg, \] from $H^p({\mathbb{R}}^n)$ to ${\dot{H}}^q({\mathbb{R}}^n)$ is comparable to $\Vert g{\Vert }_{{\dot{H}}^r({\mathbb{R}}^n)}$. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces.