Contractive projections on $H^p$-spaces

Abstract: This paper investigates contractive projections on closed subspaces $X$ of $L^p$ with $0<p<\infty$. One of the main results states that, subject to certain mild conditions, every contractive projection $P$ on $X$ preserving constants coincides with a conditional expectation on $L^\infty \cap P^{-1}(L^\infty)$. It results in some interesting applications concerning contractive idempotent coefficient multipliers for analytic function spaces and translation-invariant subspaces of $L^p(G),$ where $G$ is a compact Abelian group. Focusing specifically on descriptions of boundedness and contractivity of conditional expectations on the Hardy space $H^p(\mathbb{T})$ with $0<p<1$, we give a complete characterization of contractive idempotent coefficient multipliers for $H^p(\mathbb{T}^d)$ with $0<p<1$, which complements a remarkable result due to Brevig, Ortega-Cerd`{a}, and Seip characterizing such multipliers on $H^p(\mathbb{T}^d)$ for $1\leq p \leq \infty$.