The Operator Norm of Paraproducts on Hardy Spaces

For a tempered distribution $g$, and $0<p,q,r<\mathrm{\infty }$ with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$, we show that the operator norm of a Fourier paraproduct ${\mathrm{\Pi }}_g$, of the form $${\mathrm{\Pi }}_g(f):=\sum _{j\in \mathbb{Z}}({\phi }_{2^{-j}}\ast f)\cdot {\mathrm{\Delta }}_{j}g,$$ from $H^p({\mathbb{R}}^n)$ to ${\dot{H}}^q({\mathbb{R}}^n)$ is comparable to $\mathrm{\parallel }g{\mathrm{\parallel }}_{{\dot{H}}^r({\mathbb{R}}^n)}$. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces.