Embedding subspaces on \(L_p\) into \(\ell_p^N, 0<p<1\) (Q2747574)

Given an \(n\)-dimensional subspace \(X\) of \(L_p[0,1]\) and \(\varepsilon > 0\), what is the smallest integer \(N = N(n, \varepsilon)\) such that there is a subspace \(Y\) of \(l_p^N\) with \(d(X, Y) \leq 1 + \varepsilon\), where \(d\) is the Banach--Mazur distance? For \(p \geq 1\), the answer is known, except for possibly redundant \(\log\) factors (ignoring such factors, it is \(n\) for \(1 \leq p \leq 2\), and \(n^{p/2}\) for \(2 \leq p < \infty\)). For \(0 < p < 1\), the best known result is \(N(n, \varepsilon) \leq c(\varepsilon, p)n^2\), due to the second author [\textit{G. Schechtman}, Compos. Math. 61, 159--169 (1987; Zbl 0659.46021)]. Recently this was generalized by \textit{A.~Peña} [Math. Nachr. 189, 195--207 (1998; Zbl 0908.46011)] who showed that for any \(0 < p < 1\), any \(X\) as above admits a \(Y \subseteq l_p^N\) with \(N \leq C(p)n\) and \(d(X, Y) \leq C(p) (\log n)^{1/p}\). The problem of reducing the \(\log n\) factor to \(1 + \varepsilon\) was left open.NEWLINENEWLINENEWLINENEWLINEIn the present paper, the authors show that \(N(n, \varepsilon) \leq C(p, \varepsilon) n (\log n)^3\). Proofs for \(p \geq 1\) use Lewis' change of density result which previously was known only in that range [cf. \textit{D.~R. Lewis}, Stud. Math. 63, 207--212 (1978; Zbl 0406.46023); Mathematika 26, 18--29 (1979; Zbl 0438.46006)]. The authors extend Lewis' result to \(0 < p < \infty\) with a proof that is new even in the case \(p \geq 1\).NEWLINENEWLINETheir version of Lewis' theorem is that if \(X\) is an \(n\)-dimensional subspace of \(L_p(\Omega, \mu), 0 < p < \infty\), there is a basis \(f_1, \ldots, f_n\) of \(X\) satisfying NEWLINE\[NEWLINE \int \left( \sum_{i =1}^n f_i^2 \right)^{(p -2)/2} f_k f_l \, d\mu = \delta_kl, \quad 1 \leq k,\;l \leq n. NEWLINE\]