Hilbertian Hardy-Sobolev spaces on a half-plane
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910310353338368 |
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| author | Galé, José E. Matache, Valentin Miana, Pedro J. Sánchez--Lajusticia, Luis |
| author_facet | Galé, José E. Matache, Valentin Miana, Pedro J. Sánchez--Lajusticia, Luis |
| contents | In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel $K_{z,n}$ of $H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_φ:H_2^{(n)} \to H_2^{(n)}$, $C_φf=f\circ φ$, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps $φ: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16091 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbertian Hardy-Sobolev spaces on a half-plane Galé, José E. Matache, Valentin Miana, Pedro J. Sánchez--Lajusticia, Luis Functional Analysis Primary 46E22, 47B33, Secondary 44A10, 30H10 In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel $K_{z,n}$ of $H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_φ:H_2^{(n)} \to H_2^{(n)}$, $C_φf=f\circ φ$, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps $φ: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition operators. |
| title | Hilbertian Hardy-Sobolev spaces on a half-plane |
| topic | Functional Analysis Primary 46E22, 47B33, Secondary 44A10, 30H10 |
| url | https://arxiv.org/abs/2401.16091 |