Martingale transforms between Hardy-Orlicz spaces \(\mathcal 2_{\Phi_1}\) and \(\mathcal 2_{\Phi_2}\) of martingales (Q553014)

It is known that \(f\) is in the Hardy space \(\mathcal{D}_{p_1}\) of martingales with predictable quadratic variation if and only if \(f\) is a martingale transform of a martingale from the space \(\mathcal{D}_{p_2}\), where \(0<p_1<p_2<\infty\) (see [\textit{A. M. Garsia}, Martingale Inequalities. Seminar notes on recent progress. Reading, Mass. etc.: W. A. Benjamin, Inc. Advanced Book Program (1973; Zbl 0284.60046)] and [\textit{F. Weisz}, Martingale Hardy spaces and their applications in Fourier analysis. Berlin: Springer-Verlag (1994; Zbl 0796.60049)]). This theorem is generalized for Hardy-Orlicz spaces of martingales.