Boundedness of the Hilbert transform on Besov spaces
The Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on $L^p(\mathbb{R}^n)$ has been extensively studied by various authors in different contexts and the authors gave positive results for some or all $p,1<p<\infty$. Littlewood-Paley...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2020-12-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4057 |
| _version_ | 1827281285607849984 |
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| author | A. Maatoug S.E. Allaoui |
| author_facet | A. Maatoug S.E. Allaoui |
| author_sort | A. Maatoug |
| collection | DOAJ |
| description | The Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on $L^p(\mathbb{R}^n)$ has been extensively studied by various authors in different contexts and the authors gave positive results for some or all $p,1<p<\infty$. Littlewood-Paley theory provides alternate methods for studying singular integrals. The Hilbert transform along curves, the classical example of a singular integral operator, led to the extensive modern theory of Calderón-Zygmund operators, mostly studied on the Lebesgue $L^p$ spaces. In this paper, we will use the Littlewood-Paley theory to prove that the boundedness of the Hilbert transform along curve $\Gamma$ on Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ can be obtained by its $L^p$-boundedness, where $ s\in \mathbb{R}, p,q \in ]1,+\infty[ $, and $\Gamma(t)$ is an appropriate curve in $\mathbb{R}^n$, also, it is known that the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ are embedded into $L^p(\mathbb{R}^n)$ spaces for $s >0$ (i.e. $B^{s}_{p,q}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), s>0)$. Thus, our result may be viewed as an extension of known results to the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ for general values of $s$ in $\mathbb{R}$. |
| first_indexed | 2024-04-24T08:57:40Z |
| format | Article |
| id | doaj.art-8c0e13912399491daad1b73fc49a77ba |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:40Z |
| publishDate | 2020-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-8c0e13912399491daad1b73fc49a77ba2024-04-16T07:04:01ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102020-12-0112244345010.15330/cmp.12.2.443-4503541Boundedness of the Hilbert transform on Besov spacesA. Maatoug0https://orcid.org/0000-0002-0342-8921S.E. Allaoui1Laghouat University, 03000, Laghouat, AlgeriaLaghouat University, 03000, Laghouat, AlgeriaThe Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on $L^p(\mathbb{R}^n)$ has been extensively studied by various authors in different contexts and the authors gave positive results for some or all $p,1<p<\infty$. Littlewood-Paley theory provides alternate methods for studying singular integrals. The Hilbert transform along curves, the classical example of a singular integral operator, led to the extensive modern theory of Calderón-Zygmund operators, mostly studied on the Lebesgue $L^p$ spaces. In this paper, we will use the Littlewood-Paley theory to prove that the boundedness of the Hilbert transform along curve $\Gamma$ on Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ can be obtained by its $L^p$-boundedness, where $ s\in \mathbb{R}, p,q \in ]1,+\infty[ $, and $\Gamma(t)$ is an appropriate curve in $\mathbb{R}^n$, also, it is known that the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ are embedded into $L^p(\mathbb{R}^n)$ spaces for $s >0$ (i.e. $B^{s}_{p,q}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), s>0)$. Thus, our result may be viewed as an extension of known results to the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ for general values of $s$ in $\mathbb{R}$.https://journals.pnu.edu.ua/index.php/cmp/article/view/4057hilbert transformlittlewood-paley decompositionbesov spaces |
| spellingShingle | A. Maatoug S.E. Allaoui Boundedness of the Hilbert transform on Besov spaces Karpatsʹkì Matematičnì Publìkacìï hilbert transform littlewood-paley decomposition besov spaces |
| title | Boundedness of the Hilbert transform on Besov spaces |
| title_full | Boundedness of the Hilbert transform on Besov spaces |
| title_fullStr | Boundedness of the Hilbert transform on Besov spaces |
| title_full_unstemmed | Boundedness of the Hilbert transform on Besov spaces |
| title_short | Boundedness of the Hilbert transform on Besov spaces |
| title_sort | boundedness of the hilbert transform on besov spaces |
| topic | hilbert transform littlewood-paley decomposition besov spaces |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4057 |
| work_keys_str_mv | AT amaatoug boundednessofthehilberttransformonbesovspaces AT seallaoui boundednessofthehilberttransformonbesovspaces |