Search results for " 26B35"
showing 2 items of 2 documents
Space-filling vs. Luzin's condition (N)
2013
Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more general situations where the measures on the two metric spaces are not necessarily the same and not ne…
On Limits at Infinity of Weighted Sobolev Functions
2022
We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of…