The Operator Norm of Paraproducts on Hardy Spaces

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.