Boundedness of the Hilbert transform on Besov spaces - Maatoug, A.; Allaoui, S.E. (Carpathian Mathematical Publications, 2020)
The Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on L^p(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<∞. 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 Γ on Besov spaces B^s_p,q(R^n) can be obtained by its L^p-boundedness, where sR, p,q ]1,+∞[ , and Γ(t) is an appropriate curve in R^n, also, it is known that the Besov spaces B^s_p,q(R^n) are embedded into L^p(R^n) spaces for s >0 (i.e. B^s_p,q(R^n) L^p(R^n), s>0). Thus, our result may be viewed as an extension of known results to the Besov spaces B^s_p,q(R^n) for general values of s in R.