Journé’s covering lemma and its extension to higher dimensions

The classic example of such a T is the Hilbert transform. It is well known that operators of this type map HV(Iq) to LV(Iq) for 0 < p < o. Similarly, operators which satisfy (0.1) but with respect to differences in the x-variable are known to map L to BMO(Iq). In [4], Journ6 defines a class of Calderrn-Zygmund operators on product domains Iqn Rm and proves that such operators map L to BMO(R Iqm). Because the defining condition for BMO(Iq Rm) is not a direct generalization of the one-variable condition, the proof of this fact requires new ideas and methods. For a product domain with two factors, these ideas (of [4]) were synthesized into a geometric "coveting" lemma (also due to Journr) for rectangles in I:12. By combining this lemma with the atomic decomposition for H’(FI FIm), R. Fefferman in [2] has extended Journr’s results. There it is shown that Calder6n-Zygmund operators satisfying the product form of condition (0.1) map H(lq Fin) to L for 0 < p < 1. Our aim in this note is to