The Fourier transform of order statistics with applications to Lorentz spaces

Details The Fourier transform of order statistics with applications to Lorentz spaces The Fourier transform of order statistics with applications to Lorentz spaces

1993-11-17

arxiv.org/abs/math/9311208v1

Mathematics

Functional Analysis

Scientific paper

We present a formula for the Fourier transforms of order statistics in $\Bbb R^n$ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in $\Bbb R^n.$ For $a_1\geq ... \geq a_n\ge0$ and $q>0,$ denote by $\ell_{w,q}^n$ the $n$-dimensional Lorentz space with the norm $\|(x_1,...,x_n)\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$, where $(x_1^{*},...,x_n^{*})$ is the non-increasing permutation of the numbers $|x_1|,...,|x_n|.$ We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces into $L_q$ \cite{10} to prove that, for $n\geq 3$ and $q\leq 1,$ the space $\ell_{w,q}^n$ is isometric to a subspace of $L_q$ if and only if the numbers $a_1,...,a_n$ form an arithmetic progression. For $q>1,$ all the numbers $a_i$ must be equal so that $\ell_{w,q}^n = \ell_q^n.$ Consequently, the Lorentz function space $L_{w,q}(0,1)$ is isometric to a subspace of $L_q$ if and only if {\it either} $0

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