On the Hahn-Banach Extension Property in Hardy and mixed norm spaces on the unit ball

Abstract

For a nonempty setE of nonnegative integers letH ^{p, q, a} _E andH ^p _E be the closed linear span of

$$\left\{ {z_1^{\alpha _1 } z_2^{\alpha _2 } \ldots z_n^{\alpha _n } :(\alpha _1 ,\alpha _2 , \ldots ,\alpha _n ) \in (Z_ + )^n ,\alpha _1 + \alpha _2 + \ldots + \alpha _n \in E} \right\}$$

in the mixed norm spaceH ^{p, q, a} _E (B _n ) and in the Hardy spaceH ^p(B _n ), respectively. In this note we prove that the Hahn-Banach Extension Property (HBEP) ofH ^{p, q, a} _E is independent ofq. As an application, we show that if 0<p<1 andH ^{p, q, a} _E orH ^p _E has HBEP thenE must be thick in the sense that ifE={m_n∶n=1, 2,...}, wherem ₁<m ₂<..., thenm _n<-c n for some constantc. This result is an extension over those obtained in [2] and [4].