Minimality of the Riesz Projection Among Projections onto Abstract Hardy Spaces and Related Topics

Let X be a translation-invariant Banach function space on the unit circle T T with the associate space $$X'$$ X ′ , let w be a weight such that w∈ X w ∈ X and 1/w∈ X' 1 / w ∈ X ′ , let X ( w ) consist of measurable functions f:T→ C f : T → C such that fw∈ X f w ∈ X , and let H [ X ] and H [ X ( w )] denote the abstract Hardy spaces built upon X and X ( w ), respectively. Extending Rudin’s arguments (1962), we show that if P P is a bounded projection from X ( w ) onto H [ X ( w )], then the Riesz projection P is bounded from X onto H [ X ] and aI+bP B(X)≤ aI+bP B(X(w)) ‖ a I + b P ‖ B ( X ) ≤ ‖ a I + b P ‖ B ( X ( w ) ) for all a,b∈ C a , b ∈ C . Further, for m∈ N m ∈ N , let T(e₋ₘ) <mml:math xmlns:mml="http://www.w3.org/1998