Block version of Maurey Pisier theorem

Question

The Maurey-Pisier theorem states that if $p_X$ is the supremum of those $p$ such that the Banach space $X$ has Rademacher type $p$, then $\ell_{p_X}$ is finitely representable in $X$.

For $1\leq p<\infty$, let us say the Schauder basis $(e_i)_{i=1}^\infty$ has \emph{block type} $p$ if there exists a constant $C$ such that for any natural number $n$, any $0=k_0<\ldots <k_n$, any scalars $(a_i)_{i=1}^{k_n}$, if $y_i=\sum_{j=k_{i-1}+1}^{k_i} a_je_j$, \begin{equation}\bigl(\int_0^1 \|\sum_{i=1}^n r_i(t)y_i\|^pdt\bigr)^{1/p} \leq C \bigl(\sum_{i=1}^n \|y_i\|^p\bigr)^{1/p}.\end{equation} Here, $(r_i)_{i=1}^\infty$ is the sequence of Rademacher functions on $[0,1]$.

This definition was given on page 24 of the article "Infinite dimensional geometric moduli and type-cotype theory" by V.D. Milman and A Perelson, in the book "Geometric Aspects of Banach Spaces: Essays in Honour of Antonio Plans."

My question, which doesn't seem to be directly stated in that article, is does the block version of the Maurey-Pisier theorem hold:

$Q:$ Is it true that, if $p_X$ is the supremum of those $p\in [1,\infty)$ such that the basis $(e_i)$ has block type $p$, $\ell_{p_X}$ is block finitely representable in $(e_i)$?

The article of Milman and Perelson references "the variant of Maurey-Pisier's theorem for blocks of a given sequence as it was done, for example, in [MSch2]." However, in the bibliography of that article, there is no [MSch2] entry. I have looked through some of the bibliography items listed in that article, but I am unable to find anywhere a definitive answer to the question $Q$.

Answer 1

I believe that the answer is NO. I do not know a counterexample, however, in positive direction one can do the following.

If the basis is unconditional you can define type/cotype on disjointly supported vectors. Then the corresponding result is true. This is written here (with slightly different language), see Theorem 5.6.

The reference that you are looking for must be the paper by Milman and Sharir. A warning though, what they call block version is actually disjointly supported version (not successive blocks) i mentioned above.

Answer 2

It seems as though the answer is negative, at least for type, although I don't believe this is remarked anywhere in the literature. If $(s_i)$ is the summing basis of $c_0$, for any scalars $(a_i)_{i=1}^n$, $\|\sum_{i=1}^n a_i s_i\|\geqslant |\sum_{i=1}^n a_i|$. This means that $$\int_0^1 \|\sum_{i=1}^n r_i(t) s_i\|_{c_0} dt \geqslant \int_0^1 |\sum_{i=1}^n r_i(t)|dt \geqslant n^{1/2}/A_1,$$ where $A_1$ is the constant from Khintchine's inequality. From this it follows that $(s_i)$ couldn't have block type greater than $2$. But no $\ell_p$, $1\leqslant p<\infty$, is block finitely representable in the summing basis.