From the abstract: “Let \(A=\left( a_{n,k}\right) _{n,k\geq 0}\) be a nonnegative matrix. Denote by \(L_{l_{p}(w) ,e_{w,q}^{\theta }}(A) \) the supremum of those \(L\) satisfying the inequality \[ \left\{ \sum_{n=0}^{\infty }w_{n}\left( \sum_{k=0}^{n}\binom{n}{k}\left( 1-\Theta \right) ^{n-k_{\Theta }n}\sum_{j=0}^{\infty }a_{k,j}x_{j}\right) ^{q}\right\} ^{\frac{1}{q}}\geq L\left( \sum_{n=0}^{\infty }w_{n}x_{n}^{p}\right) ^{\frac{1}{p}}, \] where \(x\geq 0,\) \(x\in l_{p}\left( w\right) \) and \(w=( w_{n}) \) is a decreasing, non-negative sequence of real numbers.
In this paper, first we introduce the Euler weighted sequence space, \(e_{w,q}^{\theta}\) (\(0<p<1\)) of non-absolute type, which is the \(p\)-normed space included in the space \(l_{p}\left( w\right)\). Then we focus on the evaluation of \( L_{l_{p}(w) ,e_{w,q}^{\theta }}( A^{t}) \) for a lower triangular matrix \(A\), where \(0<q\leq p<1\). Also, a Hardy type formula is established for \(L_{l_{p}(w)},e_{w,q}^{\theta }( H^{t}) \), where \(H\) is Hausdorff matrix and \(0<q\leq p<1\). In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices, Cesàro matrices, Hölder matrices and Gamma matrices.”
Furthermore, the authors prove the following theorems:
Theorem 2.1. The set \(e_{w,p}^{\theta }\) becomes a linear space with the coordinatewise addition and scalar multiplication which is the \(p\)-normed space with the \(p\)-norm \[ ||| x||| :=\| x\|_{e_{w,p}^{\theta }}^{p}=\sum_{n=0}^{\infty }w_{n}\left| \sum_{k=0}^{n}\binom{n}{k}\left( 1-\Theta \right) ^{n-k}x_{k}\right| ^{p}. \] The inclusion \(e_{w,p}^{\theta }\subset l_{p}(w)\) holds for \(0<p<1\).
If \(0<s\leq r<1\), then \(e_{w,p}^{s}\subset e_{w,p}^{r}\).
Theorem 2.5. Let \(0<p<1\). Define the sequence \(b^{\left( j\right) }\left( \theta \right) =\left\{ b_{k}^{\left( j\right) }\left( \theta \right) \right\} _{k=0}^{\infty }\) of the elements of the space \(e_{w,p}^{\theta }\) by \[ b_{k}^{\left( j\right) }\left( \theta \right) =\begin{cases} 0 &\text{ if } k<j, \\ \binom{k}{j}\left( 1-\Theta \right) ^{k-j}\theta ^{-k} &\text{ if } k\geq j, \end{cases} \] for every fixed \(j=0,1,\dots\). Then the sequence \(\left\{ b^{(j)}(\theta)\right\} _{j=0}^{\infty }\) is basis for the space \( e_{w,p}^{\theta },\) and any \(x\in e_{w,p}^{\theta }\) has a unique representation of the form \[ x=\sum_{j=0}^{\infty }\left( E(\theta) x\right) _{j}b^{(j)}(\theta). \]
Theorem 3.2. Let \(0<q\leq p<1\) and \(A=\left( a_{n,k}\right) _{n,k\geq 0}\) be a lower triangular matrix with \(A\geq 0\). Then \[ L_{l_{p}( w) ,e_{w,q}^{\theta}}( A^{t}) \geq qM^{q-1}\left( \frac{1}{\theta }\right) ^{1/q}\left( \inf_{j\geq 0}\sum_{k=0}^{j}a_{j,k}\right) . \] Let \(0<q\leq p<1\), \(1/q+1/q^{\ast }=1\) and \(H_{\mu }\) be the Hausdorff matrix. Then \[ L_{l_{p}( w) ,e_{w,q}^{\theta}}( H_{\mu }^{t}) \geq \theta ^{-1/q}\int_{0}^{1}\alpha ^{1/q^{\ast }}d\mu \left( \alpha \right) . \]