A remark on the Beurling-Malliavin theorem in several variables

TL;DR

This paper extends the Beurling-Malliavin multiplier theorem to radial functions in multiple variables, simplifying previous proofs and addressing a question posed by Vasilyev regarding the Cartwright version.

Contribution

It introduces a straightforward lifting trick that generalizes the theorem to several variables and clarifies the Cartwright version, simplifying prior complex arguments.

Findings

Extension of the Beurling-Malliavin theorem to multiple variables

Simplification of Vasilyev's argument using the lifting trick

Resolution of Vasilyev's question on the Cartwright version

Abstract

We use a lifting trick to show that the Beurling-Malliavin multiplier theorem extends to radial functions in several variables in a straightforward way. This simplifies an argument of Vasilyev and also answers a question of Vasilyev on the Cartwright version of the theorem.