Given a bounded Euclidean domain Ω, we consider the sequence of optimisers of the k-th Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of Ω with fixed total volume. We show that this sequence encodes information yielding conditions for Ω to satisfy Pólya’s conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Pólya’s conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Pólya’s original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa’s bound in terms of packing density.

In this paper, we prove that every invertible 2-local or local automorphism of a simple generalized Witt algebra over any field of characteristic 0 is an automorphism. Furthermore, every 2-local or local automorphism of Witt algebras W_n is an automorphism for all n∈N. But some simple generalized Witt algebras indeed have 2-local (and local) automorphisms that are not automorphisms.

We investigate so-called Laplace–Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev– and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff–Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.

For the class of graph-directed self-similar measures on R, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.

The aim of this paper is to study the Lelong number, the integrability index and the Monge–Ampère mass at the origin of an S^1-invariant plurisubharmonic function on a balanced domain in C^n under the Schwarz symmetrization. We prove that n times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge–Ampère mass is always decreasing under the symmetrization.