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


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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 Rj 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 .

作者简介

D. Rutsky

St.Petersburg Department of the Steklov Mathematical Institute

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Email: rutsky@pdmi.ras.ru
俄罗斯联邦, St.Petersburg

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