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Let $ f $ be in $ L^1(\Omega) $ where $ \Omega $ is an open subset of $ \mathbb{R}^n $. Also, assume that $ B(x_i,r_i) $ is a collection of disjoint open balls in $ \Omega $ such that $ B(x_i,2r_i) \ …

Let $ \Omega $ be an open and bounded subset of $ \mathbb{R^n} $, and let $ f:\Omega \to \mathbb{R} $ be a continuous function. I'm looking for some (preferably, minimal) conditions on $ f $ under whi …

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly appreci …

Let $f:\Omega\subseteq\mathbb{R}^n\to\mathbb{R}^n$ be a Lipschitz function and $h$ be a vector in $\Omega$. Assume that $0\in\Omega$ and $f(0) = 0$. Also, let $\sigma:[0,1]\to\mathbb{R}^n$ be the stra …

Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. …