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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.13084 |
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| _version_ | 1866911479878385664 |
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| author | Shaabani, Shahaboddin |
| author_facet | Shaabani, Shahaboddin |
| contents | 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 $Π_g$, 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 $\|g\|_{\dot{H}^r(\mathbb{R}^n)}$. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Operator Norm of Paraproducts on Hardy Spaces Shaabani, Shahaboddin Functional Analysis 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 $Π_g$, 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 $\|g\|_{\dot{H}^r(\mathbb{R}^n)}$. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces. |
| title | The Operator Norm of Paraproducts on Hardy Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2402.13084 |