Space filling with metric measure spaces
\textit{P. Hajłasz} and \textit{J. T. Tyson} [Mich. Math. J. 56, No. 3, 687--702 (2008; Zbl 1165.28004)] proved the following version of the Hahn-Mazurkiewicz theorem: If \(Y\) is a length-compact metric space and \(n\geq2\), then there is a continuous surjection \(f:[0,1]^n\to Y\) in the Sobolev class \(W^{1,n}([0,1]^n,Y)\). In [Ill. J. Math. 53, No. 2, 365--378 (2009; Zbl 1223.46038)], \textit{K. Wildrick} and \textit{T. Zürcher} proved a version of the above result involving Sobolev-Lorentz spaces \(W^{1,n,q}([0,1]^n,Y)\), \(1<q\leq n\). In addition, they proved that the existence of a continuous surjection \(f:[0,1]^n\to Y\) in the class \(W^{1,n,1}([0,1]^n,Y)\) implies that the Hausdorff dimension of \(Y\) is at most \(n\). In the present paper, they tackle similar questions in the situation when a general metric space (the Heisenberg group, in particular) is considered instead of \([0,1]^n\).
scientific article; zbMATH DE number 6006394
scientific article