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.

We extend a recent result of Avelin, Hed, and Persson about approximation of functions f that are plurisubharmonic on a domain Ω and continuous on ˙Ω, with functions that are plurisubharmonic on (shrinking) neighborhoods of ˙Ω. We show that such approximation is possible if the boundary of Ω is C0 outside a countable exceptional set E⊂∂Ω. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous u, approximation is possible under less restrictive conditions on E. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces.

We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lpj,σj∩LN,1uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and Lp,σuloc is a “uniformly local” subspace of Lp,σloc larger than Lp,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lpj,σj∩Lquloc with q>N whenever pj<N and that, if so, the limit of f at infinity is uniform if the pj are suitably distributed. Only a few special cases have been considered in the literature, under much more restrictive assumptions that do not involve uniformly local spaces (pj=N and f vanishing at infinity, or ∂jf∈Lp∩Lq with p<N<q). Various similar results hold under integrability conditions on the higher order derivatives of f. All of them are applicable to g∗f with g∈L1 and f as above, or under weaker assumptions on f and stronger ones on g. When g is a Bessel kernel, the results are provably optimal in some cases.

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ^{$n$}and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.