Sobolev-type classes of functions with values in a metric space
The aim of the article is to define and study an analogue of Sobolev spaces \(W^1_p\) for functions with values in a metric space. Let \(\Omega\) be a domain in \(\mathbb R^n\) and let \(X\) be a complete separable metric space with metric \(d\). For a function \(f\:\Omega\to X\) and a point \(z\in X\), put \(f_z(t)=d[f(t),z]\). The author defines the Sobolev space \(W^1_p(\Omega,X) \) as the class of functions \(f\:\Omega\to X\) such that 1) the real-valued function \(f_z\) belongs to \(W^1_p(\Omega)\) for all \(z\in X\); 2) there exists a real-valued function \(w\in L_p(\Omega)\) such that \(|\nabla f_z(x)|\leq w(x)\) for almost all \(x\in\Omega\) and all \(z\in X\). The \(L^1_p\)-norm for mappings of class \(W^1_p(\Omega,X)\) is also defined and its continuity is established. The author shows that \(W^1_p(\Omega,X)\) is a complete space and proves theorems analogous to the Sobolev embedding theorems. It should be noted that different approaches to the definitions of Sobolev-type spaces were considered by \textit{N. J. Korevaar} and \textit{R. M. Schoen} in [Commun. Anal. Geom. 1, 561-659 (1993; Zbl 0862.58004)], by \textit{L. Ambrosio} in [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 439-478 (1990; Zbl 0724.49027)], by \textit{M. Biroli} and \textit{U. Mosco} in [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX Ser., Rend. Lincei, Mat. Appl. 6, No.1, 37-44 (1995; Zbl 0837.31006)], and by \textit{P. Hajłasz} in [Potential Anal. 5, No. 4, 403-415 (1996; Zbl 0859.46022)].
scientific article; zbMATH DE number 1431900