Boundedness of the Hilbert Transform on Besov Spaces.

Abstract

In this Thesis, we study the boundedness of the Hilbert transform along curves Γ(t) on Besov spaces Bsp,q(R n ), for that, we use the Littlewood-Paley theory to prove that the boundedness on Besov spaces Bsp,q(Rn ) can be obtained by its Lp(R n )-boundedness, for s ∈ R, p, q ∈ (1,∞), and Γ(t) is an appropriate curve in Rn . Furthermore, we study the boundedness on Lizorkin-Triebel spaces Fs p,q(R n ), and more than on the Localized Lizorkin-Triebel spaces (F s p,q(R n ))r with r = p, where we use the boundedness of such a transform on the spaces of vector-valued functions L p (R n ; lq ).