Continuous logic and embeddings of Lebesgue spaces

TL;DR

This paper employs continuous logic to provide new proofs of isometric embeddings between Lebesgue spaces, extending to complex spaces and introducing a novel characterization based on Banach lattices.

Contribution

It introduces a continuous logic-based approach to embed Lebesgue spaces and offers a new characterization of complex $L^p$ spaces using Banach lattices.

Findings

$L^r$ embeds into $L^p$ for $1 \\leq p \\leq r \\leq 2$

New proof using continuous logic and compactness theorem

Characterization of complex $L^p$ spaces via Banach lattices

Abstract

We use the compactness theorem of continuous logic to give a new proof that $L^r([0,1]; \mathbb{R})$ isometrically embeds into $L^p([0,1]; \mathbb{R})$ whenever $1 \leq p \leq r \leq 2$. We will also give a proof for the complex case. This will involve a new characterization of complex $L^p$ spaces based on Banach lattices.