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

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