Embedding $H^\infty(\D)$ into $L^\infty(\T)$: a proof without non-tangential limits

TL;DR

The note proves the existence of an isometric algebra embedding from $H^\infty(\mathbb{D})$ into $L^\infty(\mathbb{T})$ without invoking Fatou's theorem, by using the Poisson kernel as an approximate identity and a density argument for the Poisson kernels. It then constructs boundary data $f^*\in L^\infty(\mathbb{T})$ via weak$^*$ limits (Banach–Alaoglu) so that $f(re^{i\sigma})=\frac{1}{2\pi}\int f^*(e^{it})P_r(\sigma-t)\,dt$, yielding an isometric algebra embedding $H^\infty(\mathbb{D})\hookrightarrow L^\infty(\mathbb{T})$; a key density lemma ensures well-definedness. The approach extends to complex-valued bounded harmonic functions, and the embedding becomes surjective, giving an isometric algebra isomorphism between $h^\infty(\mathbb{D})$ and $L^\infty(\mathbb{T})$ via Poisson integration; Morera’s theorem underpins the holomorphic extensions in the argument. This framework clarifies boundary behavior and links Hardy space theory with Poisson integrals in a self-contained undergraduate-friendly setting.

Abstract

The purpose of this note is to show in an accessible and self-contained way the existence of an isometric algebra embedding from $H^\infty(\D)$ into $L^\infty(\T)$, without appealing to Fatou's classical theorem on non-tangential limits of analytic functions, and relying only on results from complex and functional analysis that are typically covered in a standard undergraduate course.

Theorems & Definitions (12)

• Lemma 1
• proof
• Remark
• Theorem 1
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['boundary-behaviour']}
• Theorem 2
• Proposition 1
• proof