Martingale Operators and Hardy Spaces with Continuous Time Generated by Them

In this paper, we introduce the martingale Hardy spaces and BMO spaces generated by an operator T in continuous time and establish the atomic decomposition theorem of the space HpT under the condition that T is predictable. We show that the BMOq spaces generated by the operator T are all equivalent and consider the sharp operator. Using the real interpolation method, we identify the interpolation spaces between the Hardy spaces and the BMO spaces. With the aid of atomic decomposition, we establish some martingale inequalities between the Hardy spaces generated by two different operators.