Boundedness conditions are found for the Hilbert transform H in Besov spaces with Muckenhoupt weights. The operator H in this situation acts on subclasses of functions from Hardy spaces. The results are obtained by representing the Hilbert transform H via Riemann–Liouville operators of fractional integration on R, on the norms of images and pre-images of which independent estimates are established in the paper. Separately, a boundedness criterion is given for the transform H in weighted Besov and Triebel–Lizorkin spaces restricted to the subclass of Schwartz functions.