Type and cotype with respect to arbitrary orthonormal systems

Abstract

Let $\on_{k \in \nz}$ be an orthonormal system on some $\sigma$-finite measure space $(\Om,p)$. We study the notion of cotype with respect to $\Phi$ for an operator $T$ between two Banach spaces $X$ and $Y$, defined by $\fco T := \inf$ $c$ such that \[ \Tfmm \pl \le \pl c \pll \gmm \hspace{.7cm}\mbox{for all}\hspace{.7cm} (x_k)\subset X \pl,\] where $(g_k)_{k\in \nz}$ is a sequence of independent and normalized gaussian variables. It is shown that this $\Phi$-cotype coincides with the usual notion of cotype $2$ iff \linebreak $\fco {I_{\lin}} \sim \sqrt{\frac{n}{\log (n+1)}}$ uniformly in $n$ iff there is a positive $\eta>0$ such that for all $n \in \nz$ one can find an orthonormal $\Psi = (\psi_l)_1^n \subset {\rm span}\{ \phi_k \p|\p k \in \nz\}$ and a sequence of disjoint measurable sets $(A_l)_1^n \subset \Om$ with \[ \int\limits_{A_l} \bet \psi_l\rag^2 d p \pl \ge \pl \eta \quad \mbox{for all}\quad l=1,...,n \pl. \] A similar result holds for the type situation. The study of type and cotype with respect to orthonormal systems of a given length provides the appropriate approach to this result. We intend to give a quite complete picture for orthonormal systems in measure space with few atoms.