Critical exponents of the Riesz projection
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913168808214528 |
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| author | Brevig, Ole Fredrik Llinares, Adrián Seip, Kristian |
| author_facet | Brevig, Ole Fredrik Llinares, Adrián Seip, Kristian |
| contents | Let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\mathbb{T}^d)$ to the Hardy space $H^p(\mathbb{T}^d)$, where $\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that $\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \leq q \leq \infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \[\mathfrak{p}_d(q) = 2+\cfrac{2}{d+\cfrac{2}{q-2}}\] for $d\geq1$ and $\frac{2d}{d+1} \leq q \leq \infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the conjecture holds in the endpoint case $q = 4/3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical exponents of the Riesz projection Brevig, Ole Fredrik Llinares, Adrián Seip, Kristian Functional Analysis Let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\mathbb{T}^d)$ to the Hardy space $H^p(\mathbb{T}^d)$, where $\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that $\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \leq q \leq \infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \[\mathfrak{p}_d(q) = 2+\cfrac{2}{d+\cfrac{2}{q-2}}\] for $d\geq1$ and $\frac{2d}{d+1} \leq q \leq \infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the conjecture holds in the endpoint case $q = 4/3$. |
| title | Critical exponents of the Riesz projection |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2402.09787 |