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Let (M,\mu) be a sigma-finite measure space. Let (T_t) be a semigroup ofpositive preserving maps on (M,\mu) with standard assumptions. We prove aH_1-BMO duality theory with assumptions only on T_t. The BMO is defined asspaces of functions f such that the L_\infty norm of sup_tT_t|f-T_tf|^2 isfinite. 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 thekernel of the semigroups of operators. This abstract argument allows to extendour main results to the noncommutative setting, e.g. the case whereL_\infty(M,\mu) is replaced by von Neuman algebras with a semifinite trace. Wealso prove a Carleson embedding theorem for semigroups of operators.

An H1-BMO duality theory for semigroups of operators