Search results for "Real line"
showing 5 items of 15 documents
Higher Order Sobolev-Type Spaces on the Real Line
2014
This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.
Tangent lines and Lipschitz differentiability spaces
2015
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space contains at least $n$ distinct tangent lines, obtained as the blow-up of $n$ Lipschitz curves, whe…
Which measures are projections of purely unrectifiable one-dimensional Hausdorff measures
2008
We give a necessary and sufficient condition for a measure p, on the real line to be an orthogonal projection of XAl for some purely 1-unrectifiable planar set A.
Hamel-isomorphic images of the unit ball
2010
In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ℝ2 and ℝ. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non-measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non-measurable (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
On one-dimensionality of metric measure spaces
2019
In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and fo…