Vol. 43, No. 2, 1972

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Some Hp spaces which are uncomplemented in Lp

Samuel Ebenstein

Vol. 43 (1972), No. 2, 327–339

Let Tj 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 Lp(Tj) denote the space of complex valued measurable functions which are integrable with respect to Haar measure on Tj. If j is finite we shall write n instead of j. The subspaces Hp(Tn) of LP(Tn), i.e. the Hardy spaces of Tn, have many well-known properties. A family of subspaces Hp(Tω) of the Lp(Tω) is defined and they are shown to have many of the same properties as the Hp(Tn). However a major difference between Hp(Tω) and Hp(Tn) is observed. If 1 < p < then Hp(Tn) is complemented in Lp(Tn), but Hp(Tω) is uncomplemented in Lp(Tω) for 1 < p < unless

p = 2.

Mathematical Subject Classification 2000
Primary: 43A70
Secondary: 46J15
Received: 12 April 1971
Revised: 13 August 1971
Published: 1 November 1972
Samuel Ebenstein