On one relaxation of the bounded-length-distortion condition in the context of metric measure spaces

We reformulate the bounded-length-distortion condition for maps between metric spaces in a certain relaxed form that requires the presence of a reference measure on the source space, which makes the new approach more natural from the perspective of maps from metric measure spaces to metric spaces. In terms of the introduced notion, we establish some mapping results in an entirely singular setting of the following general structure: a metric measure space of finite Hausdorff dimension admits a map with the relaxed bounded-length-distortion condition into a finite-dimensional normed space.