Extension and Embedding of Triebel–Lizorkin-Type Spaces Built on Ball Quasi-Banach Spaces

Abstract

Let \(\Omega \subset \mathbb {R}^n\) be a domain and X be a ball quasi-Banach function space with some extra mild assumptions. In this article, the authors establish the extension theorem about inhomogeneous X-based Triebel–Lizorkin-type spaces \(F^s_{X,q}(\Omega )\) for any \(s\in (0,1)\) and \(q\in (0,\infty )\) and prove that \(\Omega \) is an \(F^s_{X,q}(\Omega )\)-extension domain if and only if \(\Omega \) satisfies the measure density condition. The authors also establish the Sobolev embedding about \(F^s_{X,q}(\Omega )\) with an extra mild assumption, that is, X satisfies the extra \(\beta \)-doubling condition. These extension results when X is the Lebesgue space coincide with the known best ones of the fractional Sobolev space and the Triebel–Lizorkin space. Moreover, all these results have a wide range of applications and, particularly, even when they are applied, respectively, to weighted Lebesgue spaces, Morrey spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, mixed-norm Lebesgue spaces, and Lorentz spaces, the obtained results are also new. The main novelty of this article exists in that the authors use the boundedness of the Hardy–Littlewood maximal operator and the extrapolation about X to overcome those essential difficulties caused by the deficiency of the explicit expression of the norm of X.