A theorem of Littlewood and lacunary series for compact groups

Alessandro Figà-Talamanca and Daniel Rider

Vol. 16 (1966), No. 3, 505–514

DOI: 10.2140/pjm.1966.16.505

Abstract

Let G be a compact group and f ∈ L²(G). We prove that given p < ∞ there exists a unitary transformation U of L²(G) into L²(G), which commutes with left translations and such that Uf ∈ L^p. The proof is based on techniques developed by S. Helgason for a similar question. The result stated above, which is an extension of a theorem of Littlewood for the unit circle is then applied to the study of lacunary Fourier series.

Mathematical Subject Classification

Primary: 42.50

Milestones

Received: 1 April 1965

Published: 1 March 1966