Contractive projections, conditional expectations, and idempotent coefficient multipliers on Hᵖ spaces $(0<p<1)$

In this paper, we investigate contractive projections, conditional expectations, and idempotent coefficient multipliers on the Hardy spaces Hᵖ(T) for $0<p<1$. For such values of p, we first establish a general extension theorem for contractive projections in a probability Lᵖ-space. Combining this theorem with the study of conditional expectations on Hᵖ(T), we characterize a broad class of contractive projections on Hᵖ(T) that are of particular interest. Furthermore, we apply these results to give a complete characterization of contractive idempotent coefficient multipliers for the Hardy spaces Hᵖ(Tᵈ) on the d-dimensional torus for $0<p<1$ and 1≤ d≤ ∞. This complements a remarkable result of Brevig, Ortega-Cerdà, and Seip characterizing such multipliers on Hᵖ(Tᵈ) for 1≤ p ≤ ∞.