A1-Regularity and Boundedness of Riesz Transforms in Banach Lattices of Measurable Functions

Abstract

Let X be a Banach lattice of measurable functions on ℝⁿ × Ω having the Fatou property. We show that the boundedness of all Riesz transforms R_j in X is equivalent to the boundedness of the Hardy–Littlewood maximal operator M in both X and X′, and thus to the boundedness of all Calderón–Zygmund operators in X. We also prove a result for the case of operators between lattices: If Y ⊃ X is a Banach lattice with the Fatou property such that the maximal operator is bounded in Y ′, then the boundedness of all Riesz transforms from X to Y is equivalent to the boundedness of the maximal operator from X to Y , and thus to the boundedness of all Calderón–Zygmund operators from X to Y .