Let $X$ be a B-convex Banach space and let $v^1 = (v^1_1,…,v^1_n), …, v^n = (v^n_1,…,v^n_n)$ be an orthonormal basis of $\mathbb{R}^n$. My question is what one can say about $\left( \sum_i \Vert \sum_j v^i_j x_j \Vert^2 \right)^{\frac{1}{2}}$ for $x_1,…,x_n \in X$.
The type of result I am looking for is mimicking the results of Bourgain in ``A Hausdorff-Young inequality for B-convex Banach spaces’’. To be precise, I want a result of the following type: there is a function $f: \mathbb{N} \rightarrow \mathbb{R}$ such that for every B-convex Banach space $X$ there are constants $1< p\leq 2 \leq q < \infty, K \geq 1$ such that for every $x_1,…,x_n \in X$ and every orthonormal basis $v^1,…,v^n$ the following holds:
$\frac{1}{K f(n)} \left( \sum_j \Vert x_j \Vert^q \right)^{\frac{1}{q}} \leq \left( \sum_i \Vert \sum_j v^i_j x_j \Vert^2 \right)^{\frac{1}{2}} \leq K f(n) \left( \sum_j \Vert x_j \Vert^p \right)^{\frac{1}{p}},$
and $f(n)$ is grows slowly, for instance, $\lim_n \frac{f(n)}{n^\alpha} =0$ for every $\alpha >0$.
If this result is too strong for any arbitrary orthonormal system, are there known conditions on a sequence of orthonormal systems (of growing dimension) that will insure such a result?
