Optimal Embedding Theorems in Rearrangement-Invariant Banach Function Spaces: Lorentz, Orlicz, and Potential-Type Structures

Abstract

This paper develops a unified theory of potential-type embeddings in rearrangement-invariant Banach function spaces. By introducing a universal functional invariant — the embedding ratio  — we provide necessary and sufficient conditions for the continuity and compactness of embeddings between general potential spaces. The framework covers Lorentz, Orlicz, and Bessel–Riesz structures and yields explicit expressions for the optimal embedding constant . Our results unify several classical inequalities, including those of Sobolev, Hardy–Littlewood, and Adams, and extend them to variable and mixed growth contexts. The approach relies on geometric rearrangement arguments combined with potential estimates and interpolation theory. Applications to fractional and nonlinear operators are also discussed. The theory presented herein provides a new analytic perspective that integrates the geometric and functional aspects of modern potential analysis.