Two Hardy-Bennett-type theorems (Q1977422)

For a power \(p>0\), for \(0\leq c<1\) and a sequence \(\lambda:=\{\lambda_n\}\) (such that \(\lambda_n\geq 0\) for all \(n\) and \(\lambda_n>0\) for infinitely many \(n\)) define the following sequence spaces: \[ \begin{aligned} \lambda(p,c)&:= \left\{\mathbf x:\sum_{n=1}^\infty\lambda_n\Lambda_n^{-c}\left(\sum_{k=1}^n|x_k|\right)^p<\infty\right\},\\ \Lambda(p,c) &:= \left\{{\mathbf x}: \sum_{k=1}^n|x_k|^p=O(\Lambda_n^{(1-p)(1-c)})\right\},\end{aligned} \] where \(\Lambda=\{\Lambda_n\}\) and \(\Lambda_n:=\sum_{k=n}^\infty\lambda_k\). One of the main results of the paper states that if a sequence \(\mathbf x\) belongs to \(\lambda(p,c)\) then it admits a factorization \(\mathbf x= y\cdot z\) with \({\mathbf y}\in\ell^p\) and \({\mathbf z}\in\Lambda(p^*,c)\), where \(p^*\) is the conjugate power to \(p\). Conversely, if the sequence \(\lambda\) satisfies the additional condition \(\Lambda_n\leq K\Lambda_{2n}\) and the sequence \(\Lambda\) is quasi \(\beta\)-power-monotone decreasing with some positive \(\beta\), furthermore the sequence \({\mathbf x}\) admits the above factorization, then \({\mathbf x}\in\lambda(p,c)\). In the particular case \(c=0\) the results reduce to earlier ones of the author [Acta Math. Hung. 78, No. 4, 315-325 (1998; Zbl 0908.26012)]. If \(\lambda_n=n^{-p}\), \(c=0\) then the results of \textit{G.Bennett} [``Factorizing the classical inequalities'', Mem. Am. Math. Soc. 576, 1-130 (1996; Zbl 0857.26009)] can be deduced. The paper also contains factorization theorems for further sequence spaces.