Convolution operators and variable Hardy spaces on the Heisenberg group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909906533089280 |
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| author | Rocha, Pablo |
| author_facet | Rocha, Pablo |
| contents | Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq α< Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies Hölder conditions, such that $\frac{Q}{Q+N} < p_{-} \leq p(\cdot) \leq p_{+} < \frac{Q}α$. In this article we prove the $H^{p(\cdot)}(\mathbb{H}^{n}) \to L^{q(\cdot)}(\mathbb{H}^{n})$ and $H^{p(\cdot)}(\mathbb{H}^{n}) \to H^{q(\cdot)}(\mathbb{H}^{n})$ boundedness of convolution operators with kernels of type $(α, N)$ on $\mathbb{H}^{n}$, where $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \fracα{Q}$. In particular, the Riesz potential on $\mathbb{H}^{n}$ satisfies such estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convolution operators and variable Hardy spaces on the Heisenberg group Rocha, Pablo Classical Analysis and ODEs Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq α< Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies Hölder conditions, such that $\frac{Q}{Q+N} < p_{-} \leq p(\cdot) \leq p_{+} < \frac{Q}α$. In this article we prove the $H^{p(\cdot)}(\mathbb{H}^{n}) \to L^{q(\cdot)}(\mathbb{H}^{n})$ and $H^{p(\cdot)}(\mathbb{H}^{n}) \to H^{q(\cdot)}(\mathbb{H}^{n})$ boundedness of convolution operators with kernels of type $(α, N)$ on $\mathbb{H}^{n}$, where $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \fracα{Q}$. In particular, the Riesz potential on $\mathbb{H}^{n}$ satisfies such estimates. |
| title | Convolution operators and variable Hardy spaces on the Heisenberg group |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2403.11467 |