Let Ω be a Greenian domain in ℝ^{$d$},$d$≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let$u$be a numerical function on Ω, $u\not\equiv\infty$ , which is locally bounded from below. A short proof yields the following result: The function$u$is the infimum of its superharmonic majorants if and only if each set {$x$:$u$($x$)>$t$},$t$∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever$V$is a relatively compact open set in Ω and$x$∈$V$.

Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the same space.

We construct Fatou–Bieberbach domains in $\mathbb{C}^{n}$ for$n$>1 which contain a given compact set$K$and at the same time avoid a totally real affine subspace$L$of dimension <$n$, provided that$K$∪$L$is polynomially convex. By using this result, we show that the domain $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ for 1≤$k$<$n$enjoys the basic Oka property with approximation for maps from any Stein manifold of dimension <$n$.

We study the decay of approximation numbers of compact composition operators on the Dirichlet space. We give upper and lower bounds for these numbers. In particular, we improve on a result of El-Fallah, Kellay, Shabankhah and Youssfi, on the set of contact points with the unit circle of a compact symbolic composition operator acting on the Dirichlet space $\mathcal{D}$ . We extend their results in two directions: first, the contact only takes place at the point 1. Moreover, the approximation numbers of the operator can be arbitrarily subexponentially small.

It is proved that the natural embedding of a separable Banach space$X$into the corresponding Bourgain–Pisier space extends $\mathcal{L}_{\infty}$ -valued operators.