BMO-regularity in lattices of measurable functions on spaces of homogeneous type

Abstract

Let X be a lattice of measurable functions on a space of homogeneous type (S, ν) (for example, S = R n with Lebesgue measure). Suppose that X has the Fatou property. Let T be either a Calderón-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 X α L 1−α 1 β for some β ∈ (0, 1) and sufficiently small α ∈ (0, 1) if and only if X has the following simple property: for every f ∈ X there exists a majorant g ∈ X such that log g ∈ 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.

On the Relationship Between AK-Stability and BMO-Regularity

“…Corollary 13 shows that the Riesz projection is then bounded in (L 2 , Z) ζ,2 for all sufficiently small 0 < ζ < 1. Then Proposition 20 implies that Z is a BMO-regular lattice, so by the divisibility property of BMO-regularity (see, e.g., [17,Theorem 1.5]), we also have the BMO-regularity of (XY ) 1 4 , and therefore (see, e.g., [17,Theorem 5.8…”

“…For the following theorem, see [5,6,11]; the most general result of this type was established in [17]. [14]).…”

“…Then the lattice L 2 , is also BMO-regular (see, e.g.,[17, Proposition 1.11]), and by Proposition 15, this lattice is the same as the lattice L 2 l 2 , β ) (l ∞ ) , L ∞ (l ∞ )…”

“…The set A is convex and closed with respect to convergence in measure because the set BA p (C) has these properties (see, e.g., [17,Proposition 3.4]). The set A is also invariant under the action of σ l .…”

“…The set A is also invariant under the action of σ l . We fix a majorant H ∈ A and form a sequence of means H n = μ n H, where by μ n we denote the averaging operators μ n S = 1 n n−1 l=0 σ l S, n ∈ N. The convexity of the set A implies that H n ∈ A for all n ∈ N. By the Fatou property of the lattice Y (which follows from the Fatou property of X) and [17,Proposition 3.3], there exists some sequence W n = k≥n α…”