Fourier multipliers and spectral measures in Banach function spaces

Abstract

It is classical that amongst all spaces Lp (G), 1 ≤ p ≤ ∞, for \(G = {\mathbb{R}}, \mathbb{Z}\), or \({\mathbb{T}}\) say, only L2 (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L2 (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L2 (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L2 (G); this fails for every p ≠ 2. We show that this special status of L2 (G) amongst the spaces Lp (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).