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

Let Omega subset of 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-X,q(s)(Omega) for any s is an element of (0,1) and q is an element of (0, infinity) and prove that Omega is an F-X,q(s)(Omega)-extension domain if and only if Omega satisfies the measure density condition. The authors also establish the Sobolev embedding about F-X,q(s)(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.