On the span of some three valued martingale difference sequences in \(L^ p\) \((1<p<\infty)\) and \(H^ 1\) (Q1110036)

A sequence \(\{u_ n\}\) of measurable functions on a probability space is called a martingale difference sequence (m.d.s.) if for every n the conditional expectation of \(u_ n\) with respect to the algebra, generated by \(\{u_ 1,...,u_{n-1}\}\), is equal to zero. A m.d.s. \(\{u_ n\}\) is called nested if supp \(u_ i\cap \sup p u_ j\neq \emptyset\) implies that either supp \(u_ i\subset \sup p u_ j\) or supp \(u_ j\subset \sup p u_ i\). Let \(X^ p\) denote the span of a nested \(\{\)-1,0,1\(\}\)-valued m.d.s. in \(L^ p.\) The main result of the paper: the spaces \(L^ p\), \(\ell^ p\), \(\ell^ 2\), \(\ell^ p\oplus \ell^ 2\), \((\sum \oplus \ell^ 2)_ p\) are the only isomorphic types of \(X^ p\).