Optimal embeddings of the Hölder and weighted spaces of number sequences (Q1594146)

For \(p\geq 2\), a weight \({\mathcal K}= ({\mathcal K}_j)\) is defined as a number sequence such that \({\mathcal K}_1>{\mathcal K}_2>\cdots> 0\) and \(\sum^\infty_{j=1}{\mathcal K}^q_j< \infty\) with \(q:= p/(p- 1)\). The weight space \(\ell_{\mathcal K}\) is the sequence space \(E:= \{x= (x_i)\mid \sum^\infty_{i= 1}|x_i|^p< \infty\}\) with the weight norm \(\|x\|_{\mathcal K}:= \sum^\infty_{j= 1}{\mathcal K}_j|x_{i_j}|\), where \((i_j)\) is a sequence fo the subscripts \(1,2,\dots\) such that \(|x_{i_1}|\geq|x_{i_2}|\geq\cdots\). The author states (without proofs that) (1) \(\|x\|_{\mu(p)}\leq\|x\|_p\leq\|x\|_{\nu(p)}\) \((x\in E)\), where \(\|\;\|_p\) is the usual norm of \(\ell_p\), \(\mu(p)\) and \(\nu(p)\) are the weights with \(\mu_j(p):= a^{j- 1}(1- a^q)^{1/q}\) and \(\nu_j(p):= a^{j- 1}\), \(a:= 2^{1/p}- 1\), (2) the estimate of (1) cannot be improved in the class of weight norms.