On the product of functions in BMO and H¹

The point-wise product of a function of bounded mean oscillation with a function of the Hardy space H¹ is not locally integrable in general. However, in view of the duality between H¹ and BMO, we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic of the corresponding Hardy-Orlicz space can be written as a product of a function in the holomorphic Hardy space H¹ and a holomorphic function with boundary values of bounded mean oscillation.