Narrow operators are those operators defined on function spaces which are "small" at signs, i.e., at {-1,0,1}-valued functions. We summarize here some results and problems on them. One of the most interesting things is that if E has an unconditional basis then each operator on E is a sum of two narrow operators, while the sum of two narrow operators on L₁ is narrow. Recently this notion was generalized to vector lattices. This generalization explained the phenomena of sums: the set of all regular...

"Near metric" properties of the space of continuous real-valued functions on a space X with the compact-open topology or with the topology of pointwise convergence are examined. In particular, it is investigated when these spaces are stratifiable or cometrisable.

Nearly smooth points and near smoothness in Orlicz spaces are characterized. It is worth to notice that in the nonatomic case smooth points and nearly smooth points are the same, but in the sequence case they differ.

The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted $L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.

We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{\text{loc}}^{1,1}({ℝ}^N;{ℝ}^d)$ and in the space $B{V}_{\text{loc}}({ℝ}^N;{ℝ}^d)$ of functions of bounded variation.

We collect known and prove new necessary and sufficient conditions for the weighted weak type maximal inequality of the form $${\Phi }_1\left(\lambda \right)\varrho \left(\{x\in X:M_{\mu }f\left(x\right)>\lambda \}\right)\le c{\int }_X{\Phi }_2\left(c\right|f\left(x\right)\left|\right)\sigma \left(x\right)\mathrm{d}\mu \left(x\right),$$ which extends some known results.

A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_p$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_p$ (1 ≤ p < +∞).

For Tychonoff $X$ and $\alpha$ an infinite cardinal, let $\alpha defX:=$ the minimum number of $\alpha$ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $ℝ$-defect), and let $rtX:={min}_{\alpha }max(\alpha ,\alpha defX)$. Give $C\left(X\right)$ the compact-open topology. It is shown that $\tau C\left(X\right)\le n\chi C\left(X\right)\le rtX=max\left(L\right(X),L(X)defX)$, where: $\tau$ is tightness; $n\chi$ is the network character; $L\left(X\right)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C\left(X\right)=L\left(X\right)$. The (apparently new) cardinal functions $n\chi C$ and $rt$ are compared with several others.

The Nevanlinna algebras, ${}_{\alpha }^p$, of this paper are the $L^p$ variants of classical weighted area Nevanlinna classes of analytic functions on = z ∈ ℂ: |z| < 1. They are F-algebras, neither locally bounded nor locally convex, with a rich duality structure. For s = (α+2)/p, the algebra $F_s$ of analytic functions f: → ℂ such that ${(1-|z\left|\right)}^s\left|f\left(z\right)\right|\to 0$ as |z| → 1 is the Fréchet envelope of ${}_{\alpha }^p$. The corresponding algebra ${}_s^{\infty }$ of analytic f: → ℂ such that $su{p}_{z\in }{(1-|z\left|\right)}^s\left|f\left(z\right)\right|<\infty$ is a complete metric space but fails to be a topological vector space. $F_s$ is also...