Martingale transforms on Banach function spaces

We establish the boundedness of martingale transforms on Banach function spaces by using the Rubio de Francia extrapolation theory and the interpolation theorem by Zygmund. The main result also yields the boundedness of the martingale transforms on rearrangement-invariant Banach function spaces, Orlicz spaces, Lorentz-Karamata spaces, Zygmund spaces, Lebesgue spaces with variable exponents, Morrey spaces with variable exponents and Lorentz-Karamata Morrey spaces.