A characterization of BMO α -martingale spaces by fractional Car-leson measures

We give a characterization of BMO α -martingale spaces by using fractional Carleson measures. This characterization says that a martingale belongs to BMO α -martingale spaces if and only if a measure related to this martingale is a bounded α -Carleson measure. We get the boudedness of martingale transform and square function on BMO α -martingale spaces easily by using this characterization. We also prove the martingale version of Carlesons inequality related to bounded α -Carleson measures. At last, we give a characterization of UMD-space by using fractional Carleson measure.