Sobolev embeddings, extrapolations, and related inequalities

TL;DR

This paper introduces a unified approach using limiting interpolation to analyze Sobolev space embeddings across subcritical, critical, and supercritical cases, providing new characterizations and extending extrapolation theorems.

Contribution

It develops a comprehensive framework for Sobolev embeddings via limiting interpolation, unifying various inequalities and characterizations, and confirms Tao's extrapolation theorem in Sobolev contexts.

Findings

Characterization of Sobolev embeddings through pointwise inequalities

Extension of Tao's extrapolation theorem to Sobolev inequalities

Unified treatment of subcritical, critical, and supercritical cases

Abstract

In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\dot{W}^k_p(\mathcal{X}))_0, \, \mathcal{X} \in \{\mathbb{R}^d, \mathbb{T}^d, \Omega\}$, in the subcritical case ($k < d/p$), critical case ($k = d/p$) and supercritical case ($k > d/p$). We characterize the Sobolev embeddings in terms of pointwise inequalities involving rearrangements and moduli of smoothness/derivatives of functions and via extrapolation theorems for corresponding smooth function spaces. Applications include Ulyanov-Kolyada type inequalities for rearrangements, inequalities for moduli of smoothness, sharp Jawerth-Franke embeddings for Lorentz-Sobolev spaces, various characterizations of Gagliardo-Nirenberg, Trudinger, Maz'ya-Hansson-Brezis-Wainger and Brezis-Wainger embeddings, among others. In particular, we show that the Tao's extrapolation theorem holds true in the setting of Sobolev inequalities. This gives a positive answer to a question recently posed by Astashkin and Milman.