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

Abstract 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 ₁ (X) aI+b P ₁ (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-₌) <mml: math xmlns: mml="http: //www. w3. org/1998