The Fourier Transform of Order Statistics with Applications to Lorentz Spaces

We present a formula for the Fourier transforms of order statistics in Rn 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 lw,q the n-dimensional Lorentz space with the norm ‖(x1, ..., xn)‖ = (a1(x∗1) q + ...+ an(x∗n) q)1/q, where (x 1 , ..., x∗n) is the non-increasing 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 lw,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 l n w,q = l n q . 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.