$BMO, H^1$, and Calderón-Zygmund operators for non doubling measures

Abstract

Given a Radon measure µ on R d , which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space BMO(µ) when µ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming µ doubling. For instance, Calderón-Zygmund operators which are bounded on L 2 (µ) are also bounded from L ∞ (µ) into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space H 1 . Using a sharp maximal operator it is shown that operators which are bounded from L ∞ (µ) into the new BMO space and from its predual H 1 into L 1 (µ) must be bounded on L p (µ), 1 < p < ∞. From this result one can obtain a new proof of the T (1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderón-Zygmund operators bounded on L 2 (µ) with functions of the new BMO are bounded on L p (µ),

Multilinear commutators of fractional integrals over Morrey spaces with non-doubling measures

“…In 2001, Tolsa [2] developed the theory of Calderón-Zygmund operators and their commutators with RBM O functions in the setting of non-doubling measures. Later on, Chen and Sawyer [3] modified the definition of RBM O to investigate the commutators of the potential operators and RBM O functions.…”

Besov spaces with non-doubling measures

“…We first recall the definition of doubling cubes of Tolsa in [26,27]. Given α > 1 and β > α n , we say that the cube Q ⊂ R d is (α, β)-doubling if µ(αQ) ≤ βµ(Q); see [26,27] for the existence and some other basic properties of the doubling cubes.…”

“…Recently more attention has been paid to non-doubling measures. It has been shown that many results of this theory still hold without assuming the doubling property; see [16,17,18,19,23,24,25,29,5,6] for some results on Calderón-Zygmund operators, [15,26,27,28] for some other results related to the spaces BM O(µ) and H 1 (µ), and [7,8,20] for the vector-valued inequalities on the Calderón-Zygmund operators and weights. …”

Spaces of type BLO for non-doubling measures

“…For the positive Radon measure µ satisfying (1), X.Tolsa [6] has introduced the spaces RBM O(µ), which are the suitable substitutes for the classical spaces…”

“…We take notation and definitions from [6]. By a cube Q ⊂ R d we mean a closed cube centered at some point z Q ∈ supp(µ) with sides parallel to the axes.…”