Wednesday 19 March 2025, 13:30 - 14:30 in HG00.071
Martijn Caspers (TUDelft)
A noncommutative Calderón-Torchinsky theorem for \(SL(n,\mathbb R)\)
Classical Fourier multipliers are functions \( m \) on the \( n \)-dimensional real Euclidean space \( \mathbb R^n \) that act by multiplication on the Fourier domain of a function and extend boundedly to \( L^p \)-spaces \( L^p (\mathbb R^n) \). Understanding the space of Fourier multipliers is a fundamental problem in harmonic analysis. A notable highlight in the theory is the Hörmander-Mikhlin theorem that poses sharp regularity (differentiability and decay) conditions on such functions to ensure that they are Fourier multipliers. Thereafter, Calderón and Torchincky showed that these regularity conditions can be relaxed upon if \( p \) approximates 2, which is known as the Calderón-Torchinsky theorem. In the talk I will first review this result. In this talk I will show that analogue of the Calderón-Torchinsky theorem holds for \(SL(n,\mathbb R)\), the \((n \times n)\)-matrices over the reals with determinant 1, which replaces the role of \( \mathbb R^n \). In fact, it holds for a wide class of semi-simple Lie groups. The proofs are based on spherical harmonic analysis and heat kernel estimates of the Casimir operator obtained by Anker and Ji.