Contents Chapter 1. Preliminaries 8 1. The non-commutative spaces L p (M, L 2 c (Ω)) 8 2. Operator valued Hardy spaces 10 3. Operator valued BMO spaces 14 Chapter 2. The Duality between H 1 and BMO 18 1. The bounded map from2. The duality theorem of operator valued H 1 and BMO 24 3. The atomic decomposition of operator valued H 1 29 Chapter 3. The Maximal Inequality 31 1. The non-commutative Hardy-Littlewood maximal inequality 31 2. The non-commutative Lebesgue differentiation theorem andnon-tangential limit of Poisson integrals 35Chapter 4. The Duality between H p and BMO q , 1 < p < 2. 39 1. Operator valued BMO q (q > 2) 39 2. The duality theorem of H p and BMO q (1 < p < 2) 46 3. The equivalence of H q and BMO q (q > 2) 50
Abstract. We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove Lp estimates for noncommutative Riesz transforms and characterize L∞ → BMO boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderón-Zygmund theory.
Abstract. We study BMO spaces associated with semigroup of operators on noncommutative function spaces (i.e. von Neumann algebras) and apply the results to boundedness of Fourier multipliers on non-abelian discrete groups. We prove an interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 < p < ∞, with optimal constants in p.
Mathematics subject classification (2000): 46L51 (42B25 46L10 47D06)
Let T be the unit circle on R 2 . Denote by BMO(T) the classical BMO space and denote by BMOD(T) the usual dyadic BMO space on T. Then, for suitably chosen δ ∈ R, we have
We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry's results in the commutative setting. Our proof is inspired by Pisier's method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in R n (where Meyer's conjecture fails) or to the word length of free groups. Lust-Piquard's work for discrete laplacians on LCA groups is also generalized in several ways. In the context of Fourier multipliers, we will prove that Hörmander-Mihlin multipliers are Littlewood-Paley averages of our Riesz transforms. This is highly surprising in the Euclidean and (most notably) noncommutative settings. As application we provide new Sobolev/Besov type smoothness conditions. The Sobolev type condition we give refines the classical one and yields dimension free constants. Our results hold for arbitrary unimodular groups.
In this paper, we study the John-Nirenberg inequality for BMO and the atomic decomposition for H1 of noncommutative martingales. We first establish a crude version of the column (resp. row) John-Nirenberg inequality for all 0 < p < ∞. By an extreme point property of Lp-space for 0 < p ≤ 1, we then obtain a fine version of this inequality. The latter corresponds exactly to the classical John-Nirenberg inequality and enables us to obtain an exponential integrability inequality like in the classical case. These results extend and improve Junge and Musat's John-Nirenberg inequality. By duality, we obtain the corresponding q-atomic decomposition for different Hardy spaces H1 for all 1 < q ≤ ∞, which extends the 2-atomic decomposition previously obtained by Bekjan et al. Finally, we give a negative answer to a question posed by Junge and Musat about BMO.
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