A convolution metric in the space of measures and $\varepsilon$-isometries on $L_p$

A. L. Koldobskii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 151-156

Résumé

For $p>0$, $p\ne2,4,6,\dots$ we define the metric $\rho(\mu,\nu)$ on the space of measures $\mu$ on $\mathbb R$ as follows $$ \rho(\mu,\nu)=\sup_{t\in\mathbb R}|(|x|^p*\mu)(t)-(|x|^p*\nu)(t)|\cdot(1+|t|)^{-p}. $$ The Kantorovich-Rubinshtein metric $A_p(\mu,\nu)$ is defined by $$ A_p(\mu,\nu)=\inf\{\int_{\mathbb R^2}|x-y|^p\,d\psi(x,y):\pi_1\psi=\mu,\pi_2\psi=\nu\}, $$ $\pi_1, \pi_2$ being the standard projections: $\mathbb R^2\to\mathbb R^1$. Theorem 1. Let $0<p<\infty$, $p\ne2,4,6,\dots$; $\mu_n, \mu$ be probability measures on $\mathbb R$ with $\int(1+|x|^p)\,d\mu_n$. If $\lim_{n\to\infty}\rho(\mu_n,\mu)=0$ then $\lim_{n\to\infty}A_p(\mu_n,\mu)=0$. Theorem 3. Let $1\leqslant p<\infty$, $p\ne2,4,6,\dots$, $H$ be a finite dimensional subspace in $L_p([0,1])$. Then there is a continuous function $\tau_H(\varepsilon)$ on $[0,\infty)$ such that $\lim_{\varepsilon\to0}\tau_H(\varepsilon)=0$ and for every linear $\varepsilon$-isometric operator $T\colon H\to L_p([0,1])$ there exists a linear isometry $U\colon H\to L_p([0,1])$ such that $\|T-U\|<\tau_H(\varepsilon)$.