Between the enhanced power graph and the commuting graph

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of $G$ act as automorphisms of the deep commuting graph.     

Atomic decomposition for preduals of Sobolev multiplier spaces

We characterize the preduals of Sobolev multiplier spaces by means of atomic decomposition. It will be shown that the vanishing integral condition plays a main role in the atomic decomposition. Certain subspaces of the preduals which consist of functions with vanishing integrals are also studied.