BMO-REGULARITY IN LATTICES OF MEASURABLE FUNCTIONS ON SPACES OF HOMOGENEOUS TYPE

Let X be a lattice of measurable functions on a space of homogeneous type (S, nu) (for example, S = R-n with Lebesgue measure). Suppose that X has the Fatou property. Let T be either a Calderon Zygmund singular integral operator with a singularity nondegenerate in a certain sense, or the Hardy-Littlewood maximal operator. It is proved that T is bounded on the lattice ((XL11-alpha)-L-alpha)(beta) some beta is an element of (0,1) and sufficiently small a alpha is an element of (0, 1) if and only if X has the following simple property: for every f is an element of X there exists a majorant g is an element of X such that log g is an element of BMO with proper control on the norms. This property is called BMO-regularity. For the reader's convenience, a self-contained exposition of the BMO-regularity theory is developed in the new generality, as well as some refinements of the main results.