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While I do not know much about current development of the general theory of distributions, I can say something about the current research topics in a special class of distributions, the theory of Sobo …

That is true. Caccioppoli sets are also known as sets of finite perimeter. Theorem. Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the fu …

Definition. If $U\subset\mathbb{R}$ is open, we say that $u\in {AC}(U)$ if $u$ is absolutely continuous on every compact interval in $U$. Let $\Omega\subset\mathbb{R}^n$. We say that $u$ is absolutely …

I think the best treatment of basic facts about capacity from the perspective of Sobolev spaces is in Chapter 4 of L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised …

The density result is true for any family of vector fields with Lipschitz coefficients. Theorem. Let $X_1,\ldots,X_k$ be a system of vector fields with Lipschitz coefficient on a compact Riemanni …

Any function in $W^{1,p}$, $p>N$, has a continuous representative by the Sobolev embedding theorem so there is no issue here. However: Proposition. There is a function $f\in W^{1,p}$, $p\leq N$, …

There are plenty of examples of discontinuous Sobolev function in $W^{1,n}(\mathbb{R}^n)$. For example $f(x)=\log|\log|x||$ defined in a neighborhood of zero. Now take $n=2$ and restrict the function …

If the annulus is small, then it is basically an Euclidean annulus and there is an extension operator for Sobolev spaces. However, if the annulus is large is may happen that it goes around a "neck" in …

Example 7.7 in L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.

This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf which I state below. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 …

The following result is due to Asplund [A, p.235]. Theorem. If $\varnothing\neq E\subset\mathbb{R}^n$ is closed, and $d(x)=\operatorname{dist}(x,E)$, then $f:\mathbb{R}^n\to\mathbb{R}$ defined by $f( …

Yes if you choose a suitable representative of a Sobolev function. Lemma. Let $f\in W^{1,p}(\mathbb{R}^n)$, $1\leq p<\infty$. Then for every $\epsilon>0$, there is a Lipschitz function $g:\mathbb{R}^n …

You actually do not need to assume that the mappings are Lipschitz as it is true for general $W^{1,n}$ mappings Theorem. If $\mathcal{M}$ and $\mathcal{N}$ are smooth compact and oriented manifolds, …

The lemma is due to: J. O. Strömberg, and A. Torchinsky. Weights, sharp maximal functions and Hardy spaces. Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1053–1056. The lemma is stated there with …

The property (P) indeed characterizes the Sobolev space $W^{1,p}$. Theorem 1. $f\in W^{1,p}(\mathbb{R}^n)$, $1<p\leq\infty$ if and only if $f\in L^p$ and there is $0\leq g\in L^p$ such that $$ | …