Global Stein Theorem on Hardy Spaces

Abstract

Let f be an integrable function which has integral 0 on ℝn. What is the largest condition on ❘ f ❘ that guarantees that f is in the Hardy space H1 (ℝn)? When f is compactly supported, it is well-known that the largest condition on ❘f❘ is the fact that ❘f❘ ∈ L log L(ℝn). We consider the same kind of problem here, but without any condition on the support. We do so for H1 (ℝn), as well as for the Hardy space Hlog (ℝn) which appears in the study of pointwise products of functions in H1 (ℝn) and in its dual BMO.