文摘
For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator S : lpn ? \mathbbRN{S : \ell_p^n \to \mathbb{R}^N} such that for every 0 < r < p < 2 with r ≤ 1, the operator Sr = S : lpn ? lrN{S_r = S : \ell_p^n \to \ell_r^N} satisfies with overwhelming probability that ||Sr|| ||(Sr)| Im S-1|| £ C(p,r)n/(N-n){\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}, where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.