# Asymptotic geometric analysis, harmonic analysis and related by McIntosh A., Portal P. (eds.) PDF

By McIntosh A., Portal P. (eds.)

ISBN-10: 0731552067

ISBN-13: 9780731552061

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Example text

This result of Bourgain’s [8] had apparently no scalar-valued predecessor, but the scalar version of the theorem was independently discovered around the same time by Yamazaki [60]. The original statement in [8] concerns n = 1, but the transference to n > 1 is standard and may be found from [26]. 1 starts from the defining inequality of UMD spaces, but now applied to a less obvious choice of the filtration. Every second σ-algebra will be generated by simple intervals N −1 [k, k + 1[, k ∈ Z, but the atoms of the intermediate σ-algebras will be unions N −1 [k, k + 1[ ∪ [n + k, n + k + 1[ , k ≡ 0, 1, · · · , n − 1 (mod 2n), of two separated intervals, where the separation reflects the translations that we are aiming at.

To establish the identity κ(B1 (0)) = Ln+1 , suppose that η = 0. 5), the set γC (ζ) is an (n − 1)-dimensional ball or sphere with radius |η| in Rn+1 , lying in the hyperplane with normal η and passing through ξ ∈ Rn+1 . |η| cos(∠(ξ, η)). The projection of ξ onto {η}⊥ has length |ξ| sin(∠(ξ, η)) and the projection of ξ onto η has length |ξ| cos(∠(ξ, η)). The projection of γC (ζ) onto {η}⊥ is a ball or sphere whose maximum distance from the origin is |ξ| sin(∠(ξ, η)) + |η| in the direction of the projection of ξ onto {η}⊥ .

Martingales and Fourier analysis in Banach spaces. , 1206, Springer, Berlin, 1986. [13] Burkholder, D. , Martingales and singular integrals in Banach spaces. Handbook of the geometry of Banach spaces, Vol. I, 233–269, NorthHolland, Amsterdam, 2001. , Sukochev, F. , Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), no. 2, 135–163. , An operator-valued transference principle and maximal regularity on vector-valued Lp -spaces. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 67–87, Lecture Notes in Pure and Appl.