An H1-BMO duality theory for semigroups of operators

Let (M,\mu) be a sigma-finite measure space. Let (T_t) be a semigroup of positive preserving maps on (M,\mu) with standard assumptions. We prove a H_1-BMO duality theory with assumptions only on T_t. The BMO is defined as spaces of functions f such that the L_\infty norm of sup_tT_t|f-T_tf|^2 is finite. The H1 is defined by square functions of P. A. Meyer's gradient form. Our argument does not rely on any geometric/metric structure of M nor on the kernel of the semigroups of operators. This abstract argument allows to extend our main results to the noncommutative setting, e.g. the case where L_\infty(M,\mu) is replaced by von Neuman algebras with a semifinite trace. We also prove a Carleson embedding theorem for semigroups of operators.