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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2408.08366 |
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- It is shown that for $0<p,q,r<\infty$, with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$, the operator norm of the dyadic paraproduct of the form \[ π_g(f) := \sum_{R \in \mathcal{D}\otimes\mathcal{D}} g_R \left\langle f \right\rangle_{R} h_R, \] from the bi-parameter dyadic Hardy space $H_d^p(\mathbb{R}\otimes\mathbb{R})$ to $\dot{H}_d^q(\mathbb{R}\otimes\mathbb{R})$ is comparable to $\|g\|_{\dot{H}_d^r(\mathbb{R}\otimes\mathbb{R})}$. We also prove that for all $0 < p < \infty$, there holds \[ \|g\|_{BMO_d(\mathbb{R}\otimes\mathbb{R})} \simeq \|π_g\|_{H_d^p(\mathbb{R}\otimes\mathbb{R}) \to \dot{H}_d^p(\mathbb{R}\otimes\mathbb{R})}. \] Similar results are obtained for bi-parameter Fourier paraproducts of the same form.