On the product of functions in H1 and BMO

Aline Bonami,

Tadeusz Iwaniec,

Peter Jones,

Michel Zinsmeister

The space BMO:

The Hardy space H1

Fefferman-Stein: BMO is the dual of H1

But this duality is not like Lp-Lq

i.e. bh need not be integrable if b is in BMO and h is in H1

Two (equivalent) ways to define the duality

What can be said about this distribution?

The answer involves the notion of Orlicz space

This theorem has a converse in the case of the disc, in the holomorphic setting:

Idea of proofs:

Proof of the theorem about holomorphic Hardy spaces