TY - JOUR AU - Maatoug, A. AU - Allaoui, S.E. PY - 2020/12/29 Y2 - 2024/03/29 TI - Boundedness of the Hilbert transform on Besov spaces: Array JF - Carpathian Mathematical Publications JA - Carpathian Math. Publ. VL - 12 IS - 2 SE - Scientific articles DO - 10.15330/cmp.12.2.443-450 UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/4057 SP - 443-450 AB - <p>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 $p,1&lt;p&lt;\infty$. 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 $s &gt;0$ (i.e. $B^{s}_{p,q}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), s&gt;0)$. 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 $\mathbb{R}$.</p> ER -