Sobolev embeddings, extrapolations, and related inequalities

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.