Optimal embeddings of critical Sobolev–Lorentz–Zygmund spaces
Volume 223 / 2014
Studia Mathematica 223 (2014), 77-95
MSC: Primary 46E35; Secondary 26D10.
DOI: 10.4064/sm223-1-5
Abstract
We establish the embedding of the critical Sobolev–Lorentz–Zygmund space $H^{{n}/{p}}_{p,q,\lambda _1,\ldots ,\lambda _m}(\mathbb R^n)$ into the generalized Morrey space ${\cal M}_{\varPhi ,r}(\mathbb R^n)$ with an optimal Young function $\varPhi $. As an application, we obtain the almost Lipschitz continuity for functions in $H^{{n}/{p}+1}_{p,q,\lambda _1,\ldots ,\lambda _m}(\mathbb R^n)$. O'Neil's inequality and its reverse play an essential role in the proofs of the main theorems.
