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The Fourier transform of order statistics with applications to Lorentz spaces
 Israel Math. J
, 1995
"... Abstract. We present a formula for the Fourier transforms of order statistics in R n showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in Rn. For a1 ≥... ≥ an ≥ 0 and q> 0, denote by ℓn w,q the ndimensional Lorentz space with the norm ‖(x ..."
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Abstract. We present a formula for the Fourier transforms of order statistics in R n showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in Rn. For a1 ≥... ≥ an ≥ 0 and q> 0, denote by ℓn w,q the ndimensional Lorentz space with the norm ‖(x1,..., xn) ‖ = (a1(x ∗ 1)q +... + an(x ∗ n)q) 1/q, where (x ∗ 1,...,x∗n) is the nonincreasing permutation of the numbers x1,..., xn. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces into Lq [10] to prove that, for n ≥ 3 and q ≤ 1, the space ℓn w,q is isometric to a subspace of Lq if and only if the numbers a1,...,an form an arithmetic progression. For q> 1, all the numbers ai must be equal so that ℓn w,q = ℓnq. Consequently, the Lorentz function space Lw,q(0, 1) is isometric to a subspace of Lq if and only if either 0 < q < ∞ and the weight w is a constant function (so that Lw,q = Lq), or q ≤ 1 and w(t) is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions. 1.
GATEAUX DIFFERENTIABILITY FOR FUNCTIONALS OF TYPE ORLICZLORENTZ
"... Abstract. Let (Ω,A, µ) be a σfinite nonatomic measure space and let Λw,φ be the OrliczLorentz space. We study the Gateaux differentiability of the functional Ψw,φ(f) = 0 φ(f∗)w. More precisely we give an exact characterization of those points in the OrliczLorentz space Λw,φ where the Gateaux deri ..."
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Abstract. Let (Ω,A, µ) be a σfinite nonatomic measure space and let Λw,φ be the OrliczLorentz space. We study the Gateaux differentiability of the functional Ψw,φ(f) = 0 φ(f∗)w. More precisely we give an exact characterization of those points in the OrliczLorentz space Λw,φ where the Gateaux derivative exists. This paper extends known results already on Lorent spaces, Lw,q, 1 < q <∞. The case q = 1, it has been considered. 1.