The Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi$, with $\phi$ a function of positive lower type and upper type less than $1$, were defined in [HV2] in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of order $\alpha$, where $\phi \left(t\right)=t^{\alpha }$, given in [GSV]. In this work we show that the composition $T_{\phi }=D_{\phi }\circ I_{\phi }$ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of $I_{\phi }$ and $D_{\phi }$ or the...

Jachymski showed that the set $$\left\{(x,y)\in {𝐜}_0\times {𝐜}_0:{\left(\sum _{i=1}^n\alpha \left(i\right)x\left(i\right)y\left(i\right)\right)}_{n=1}^{\infty }\text{is}\text{bounded}\right\}$$ is either a meager subset of ${𝐜}_0\times {𝐜}_0$ or is equal to ${𝐜}_0\times {𝐜}_0$. In the paper we generalize this result by considering more general spaces than ${𝐜}_0$, namely ${𝐂}_0\left(X\right)$, the space of all continuous functions which vanish at infinity, and ${𝐂}_b\left(X\right)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.

If $(\Omega ,\Sigma )$ is a measurable space and $X$ a Banach space, we provide sufficient conditions on $\Sigma$ and $X$ in order to guarantee that $\mathrm{b}vca(\Sigma ,X)$, the Banach space of all $X$-valued countably additive measures of bounded variation equipped with the variation norm, contains a copy of $c_0$ if and only if $X$ does.

Let $A={\left(a_{n,k}\right)}_{n,k\ge 1}$ be a non-negative matrix. Denote by $L_{v,p,q,F}\left(A\right)$ the supremum of those $L$ that satisfy the inequality $${\parallel Ax\parallel }_{v,q,F}\ge L{\parallel x\parallel }_{v,p,F},$$ where $x\ge 0$ and $x\in l_p(v,F)$ and also $v={\left(v_n\right)}_{n=1}^{\infty }$ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_{v,p,F}\left(A\right)$ instead of $L_{v,p,p,F}\left(A\right)$. In this paper we obtain a Hardy type formula for $L_{v,p,q,F}\left(H_{\mu }\right)$, where $H_{\mu }$ is a Hausdorff matrix and $0<q\le p\le 1$. Another purpose of this paper is to establish a lower bound for $\parallel A_W^{NM}{\parallel }_{v,p,F}$, where $A_W^{NM}$ is the Nörlund matrix associated with the sequence $W={\left\{w_n\right\}}_{n=1}^{\infty }$ and $1<p<\infty$. Our results generalize some works of Bennett, Jameson and...