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Sobolev classes of Banach space-valued functions and quasiconformal mappings. (English) Zbl 1013.46023

A triple \((X,d,\mu)\) is called a metric measure space if \((X,d)\) is a metric space and \(u\) is a complete, regular Borel measure on \(X\) for which \(\mu(B(a,r)) > 0\) for each \(a\in X\), \(r>0\). If \(\mu\) has finite total mass, then for each Banach space \(V\) and \(1\leq p <\infty\) the Sobolev space \(N^{1,p}(X:V)\) is introduced. The elements of \(N^{1,p}(X:V)\) are classes of functions on \(X\) with values in \(V\) which are uniquely defined on \(X\) up to a set of \(p\)-capacity zero. They get characterized by various properties.
Further, Poincaré inequalities for certain classes of Banach space valued functions on \(X\) are considered and it is shown that they do not depend on the Banach space if \(\mu\) is doubling, i.e., there exists \(C\geq 1\) such that \(\mu(B(a,2r))\leq C\mu(B(a,r))\) for all \(a\in X\), \(r > 0\).
Among other things, \(N^{1,p}(X : V)\) is compared with the Sobolev spaces considered by Korevaar-Schoen and the role of quasisymmetric embeddings \(F : X \to V\) and the quasiconformal mappings \((X,d)\to (X',d)\) in connection with Sobolev spaces is investigated.

MSC:

46E40 Spaces of vector- and operator-valued functions
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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