Atomic Hardy space theory for unbounded singular integrals

Abstract

T f (x) = lim ε→0 |y|≥ε B(y) y f (x − y) dy where the function B is non-negative and even, and is allowed to have singularities at zero and infinity. The operators we consider are not generally bounded on L 2 (R), yet there is a Hardy space theory for them. For each T there are associated atomic Hardy spaces, called H 1 B and H 1,1 B. The atoms of both spaces possess a size condition involving B. The operator T maps H 1,1 B and certain H 1 B continuously into H 1 ⊂ L 1. The dual of H 1 B is a space we call BMO B. The Hilbert transform is a special case of an operator T and its H 1 B and BMO B spaces are H 1 and BMO.