Maximal operators and B.M.O. for Banach lattices

Abstract

We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = oo. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMO X . This turns out to depend strongly on the convexity of the Banach lattice X. We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.1991 Mathematics subject classification: 42A28, 46B20, 43A25. IntroductionIn our paper [8] we have introduced the Hardy-Littlewood property (sometimes abbreviated as H.L. property) of a Banach lattice and have characterized it by several boundedness properties of the lattice version M of the Hardy-Littlewood maximal operator or its smooth variants. We refer the reader to this paper and also to [12] for the basic notation and terminology concerning Banach lattices and function spaces.As we explain in [8], the Hardy-Littlewood property is intimately connected to the U.M.D. property, which plays a central role in the development of Vector-Valued Fourier Analysis and, in spite of having been extensively studied (see [6] and [3]), still has some aspects that need to be completely understood. By studying systematically the Hardy-Littlewood property, we hoped to cast some light on the behaviour of Banach lattices regarding the U.M.D. property.The key idea in [8] is that the maximal operator M has a smooth version, obtained by replacing the averages over balls by the convolutions with an appropriate approximate identity. This smooth version can be viewed as a vector-valued singular integral, and its boundedness properties are a consequence of the general theory which was originally set up in [1] and then fully developed in [14]. This latter paper will also be a basic reference for the present work. In particular, the notation for all our function spaces will be taken from it.When one is interested in estimates on the Bochner-Lebesgue spaces L X (K"), only the size matters, and it is irrelevant whether we look at M or to its smooth version

Sparse domination for the lattice Hardy–Littlewood maximal operator

“…• M µ D can be viewed as a vector-valued singular integral (see [9,10]). The sparse domination for vector-valued singular integrals follows by combining [15, Theorem 2.10] (dominating vector-valued singular integrals by more complex operators) and [5, Theorem A] (dominating the more complex operators by the sparse operator A µ 1,S ).…”

“…Strong L p -bound implies weak L 1 -bound As well-known, for the dyadic lattice Hardy Littlewood maximal operator the strong L p -boundedness implies the weak L 1 -boundedness. This result can be proven by viewing the lattice maximal operator as a vector-valued singular integral operator (see [9,10]) and using the Calderón-Zygmund decomposition, or alternatively, by viewing the lattice maximal operator as a martingale transform (see [29]) and using the Gundy decomposition. In this Appendix, we give an elementary proof of this result.…”

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

“…Let F be the stopping family defined by the following stopping children: For each F ∈ F , the children ch F (F ) are the maximal dyadic cubes F ′ ⊆ F such that (A. 5) sup…”

Recent Developments in Real and Harmonic Analysis

“…Several characterizations and results concerning this property were achieved in [GMT,GMT2]. I would like to point out here that Carlos Segovia became interested in the interplay between the geometry of Banach spaces and vector-valued analysis, and we refer the reader to his work in [HMST] for a combination of techniques from A p theory, harmonic analysis, and the use of the Hardy-Littlewood property in Banach lattices.…”