Subspaces of \(L_ p\), \(p > 2\), with unconditional bases have equivalent partition and weight norms (Q818727)

For a sequence of real scalars \((a_i)\), a norm determined by partitions and weights is defined as follows. For a partition \(P = \{N_i\}\) of \(\mathbb N\) and a weight function \(W: \mathbb N \to (0,1]\), set \[ \bigl\| (a_i) \bigr\| _{P,W} = \Bigl( \sum\limits_i \Bigl( \sum\limits_j a_j^2 \bigl(W(j)\bigr)^2 \Bigr)^\frac{p}{2} \Bigr)^\frac{1}{p}. \] Given a family \({\mathcal P} = (P_k, W_k)\) of pairs of partitions and weight functions, the value \(\bigl\| (a_i) \bigr\| _{\mathcal P} = \sup\limits_k \bigl\| (a_i) \bigr\| _{P_k,W_k}\) is called a norm determined by partitions and weights [\textit{D.~E.\ Alspach} and \textit{S.--M.\ Tong}, Stud.\ Math.\ 159, No.~2, 207--227 (2003; Zbl 1057.46014)]. The main result asserts that any subspace of \(L_p\), \(2<p<\infty\), with an unconditional basis has an equivalent norm determined by partitions and weights, answering, in particular, a question of the authors raised in [op.cit.].