A Banach lattice E is called p-concave, \(1\leq p0\), \(\sum^{\infty}_{n=0}b_ n=1\), \(mc_{n,k}=2^{-n}b_ n\) and the sets \(\{c_{n,k}\}\) are disjoint. II. Suppose E is a symmetric space and \(1/20\). There exists a constant \(M(p,\epsilon)>0\) such that \[ \frac{1}{M(p,\epsilon)}(\sum_{(n,k)\in \Omega,n\geq 1}| c_{n,k}|^ p2^{-n}n^{-1+p/2- \epsilon})^{1/p}\leq \| \sum_{(n,k)\in \Omega}c_{n,k}\chi^ k_ n\|_{L_ p}\leq (\sum_{(n,k)\in \Omega}| c_{n,k}|^ p2^{-n})^{1/p}. \] Let G denote the closure of the set of bounded measurable functions on [0,1] with respect to the norms \(\| x\|_ G=\inf \{\lambda >0:\quad \int^{1}_{0}(e^{(x(t)/\lambda)^ 2}- 1)dt\leq 1\}.\) IV. Suppose E is a symmetric space. For the norms \(\| \sum_{(n,k)\in \Omega}c_{n,k}\chi^ k_ n\|_ E\) and \((\sum_{(n,k)\in \Omega}c^ 2_{n,k}2^{-n})^{1/2}\) to be equivalent on the set of sequences \(\{c_{n,k}\}\) satisfying the condition \(| c_{n,k}| \leq A| c_{n-1,m}|\) \((1\leq k\leq 2^ n\), \(1\leq m\leq 2^{n-1})\) for some \(A>0\) it is necessary and sufficient that \(E\supset G\).