Some H^p spaces which are uncomplemented in L^p

Samuel Ebenstein

Vol. 43 (1972), No. 2, 327–339

DOI: 10.2140/pjm.1972.43.327

Abstract

Let T^j denote the compact group which is the Cartesian product of j copies of the circle where j is a positive integer or ω. If 1 ≦ p ≦∞ let L^p(T^j) denote the space of complex valued measurable functions which are integrable with respect to Haar measure on T^j. If j is finite we shall write n instead of j. The subspaces H^p(Tⁿ) of LP(Tⁿ), i.e. the Hardy spaces of Tⁿ, have many well-known properties. A family of subspaces H^p(T^ω) of the L^p(T^ω) is defined and they are shown to have many of the same properties as the H^p(Tⁿ). However a major difference between H^p(T^ω) and H^p(Tⁿ) is observed. If 1 < p < ∞ then H^p(Tⁿ) is complemented in L^p(Tⁿ), but H^p(T^ω) is uncomplemented in L^p(T^ω) for 1 < p < ∞ unless

Mathematical Subject Classification 2000

Primary: 43A70

Secondary: 46J15

Milestones

Received: 12 April 1971

Revised: 13 August 1971

Published: 1 November 1972