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The dual of $\mathrm{BV}$

$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
Gary Moon's user avatar
  • 683
6 votes
0 answers
271 views

Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that \begin{equation} \sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
Paz's user avatar
  • 61
6 votes
0 answers
239 views

Sheaves on Rectifiable Sets

Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory? Background: The jumping off point here is a simple analogy - geometric measure ...
Juan Sebastian Lozano's user avatar
6 votes
1 answer
897 views

Flat norm metrizes the weak* topology

I've come across the following statement in literature (without proof or reference) about the flat norm of currents $$ F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...
golden-rabbit's user avatar
5 votes
0 answers
198 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
Riku's user avatar
  • 839
4 votes
0 answers
213 views

Classification of Euclidean-invariant measures?

Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely, By ...
Tim Campion's user avatar
  • 63.9k
4 votes
0 answers
99 views

Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?

Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces. Fix $\alpha>0$ and define the space, analogously to this paper, as the collection of all measurable functions $f:X\...
ABIM's user avatar
  • 5,405
4 votes
0 answers
115 views

Box counting dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it? The analogous question for the ...
Riku's user avatar
  • 839
4 votes
0 answers
123 views

Converse on the rectifiability of products of rectifiable sets

Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that : (1) $E$ is $k$ rectifiable if there exists $C\...
Paul-Benjamin's user avatar
3 votes
0 answers
222 views

Sets of finite perimeter: intersection with an half space

I have a question regarding sets of finite perimeter. In particular I'm interested to find $$\mu_{E \cap H_t}, \label{1}\tag{1}$$ where $E$ is a set of finite perimeter in a generic open set $\Omega \...
ty88's user avatar
  • 51
3 votes
0 answers
73 views

"Almost" absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$

Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...
Riku's user avatar
  • 839
3 votes
0 answers
229 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced $\...
Turbo's user avatar
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3 votes
0 answers
860 views

decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
Mikhail Katz's user avatar
  • 16.6k
3 votes
0 answers
1k views

weak regularity conditions for regions to assure boundary of measure zero

Let $\Omega \subset \mathbb{R}^d$ be a region ( bounded, simply connected, open set ). What are some regularity conditions to assure the boundary $\partial\Omega$ is a set of (lebesgue-)measure zero? ...
Alexander Thumm's user avatar
2 votes
0 answers
29 views

Steiner symmetrization of smooth function on non-simply connected regions

Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
MathLearner's user avatar
2 votes
0 answers
202 views

Prove or disprove that $u=0$ a.e. on $\Bbb R^d$

Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
82 views

Estimate of Wasserstein distance and flow of vector fields under particular assumptions

Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow. A classical estimate ...
Jun's user avatar
  • 303
2 votes
0 answers
186 views

Metric on space of Borel-measurable functions

Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
Bernard_Karkanidis's user avatar
2 votes
0 answers
92 views

First Dirichlet eigenvalue below second Neumann eigenvalue?

Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary. I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
Landauer's user avatar
  • 173
2 votes
0 answers
90 views

Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces

In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
71 views

Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?

Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following: Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
Riku's user avatar
  • 839
2 votes
0 answers
73 views

Alberti rank-one theorem and irregular jump discontinuities

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
Riku's user avatar
  • 839
2 votes
0 answers
73 views

Projection of BV function

Let $u \in [BV(\mathbb R^N)]^N$. We have $$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$ where $a,b \in \mathbb S^{N-1}$. What is the projection of $D^{jump}u$ in the direction $a$? And how can ...
Riku's user avatar
  • 839
2 votes
0 answers
199 views

Convergence of the difference quotient of a BV function

Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$. What can be said about the difference quotient $$ \frac{u(x+\epsilon y)-u(x)}{\epsilon} $$ regarding its convergence as $\epsilon \to 0$...
Riku's user avatar
  • 839
2 votes
0 answers
165 views

Jacobian and Jacobian matrix of solutions of ODE with Sobolev vector field

Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{...
user avatar
2 votes
0 answers
279 views

Relationship between $p$-capacity and Riesz $s$-capacity of a set

What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
Riku's user avatar
  • 839
2 votes
0 answers
159 views

What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?

Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...
XIII's user avatar
  • 747
2 votes
0 answers
82 views

Volume of critical points decreases under symmetric decreasing rearrangement

In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...
big_huong's user avatar
1 vote
0 answers
72 views

Initial-boundary value problem for transport equation with $W^{1,p}$ velocity

Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation $$ \begin{cases} u_t + v(t,x) u_x = 0 \qquad & (...
user175203's user avatar
1 vote
0 answers
141 views

On the uniform boundedness principle and the space of functions of bounded variation

Let $U$ be a bounded smooth domain of $\mathbb{R}^d$. We write $m$ for the Lebesgue measure on $U$. A function $f \in L^1(U,m)$ has bounded variation in $U$ if \begin{align*} V(f,U):=\sup \left\{\int_{...
sharpe's user avatar
  • 721
1 vote
0 answers
45 views

Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way: Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
Hheepp's user avatar
  • 371
1 vote
0 answers
100 views

Weak estimate for difference quotient of BV function

In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
Riku's user avatar
  • 839
1 vote
0 answers
88 views

If $u$ is $BV$ then $\operatorname{curl} Du = 0$ in the sense of distributions

Let $u\in BV(\mathbb{R}^N; \mathbb{R}^M)$. How does one prove that $$\operatorname{curl} Du = 0$$ holds in the sense of distributions?
Riku's user avatar
  • 839
1 vote
0 answers
47 views

Consistency of the definition of total variation for functions of one or several variables

Where can I find a proof that the definition of total variation for functions of several variables is consistent with the definition of total variation for functions of one variable?
Riku's user avatar
  • 839
1 vote
0 answers
107 views

Level sets of a BV function and its derivative

Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$? More specifically, does Alberti ...
Riku's user avatar
  • 839
1 vote
0 answers
92 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user avatar
1 vote
0 answers
259 views

Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?

The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
gradstudent's user avatar
  • 2,246
0 votes
0 answers
94 views

When can an affine functional on the dual be represented as an element of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
i like math's user avatar
0 votes
0 answers
92 views

Finding set of best approximations from a point in $c_0$ to its subspace

Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
PPB's user avatar
  • 85
0 votes
0 answers
43 views

Minimal condition on set for an optimisation problem

We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem: $$ \sup\{ \int_{E}(\...
JaberEdgar's user avatar
0 votes
0 answers
63 views

Coarea-like formula for BV functions (not their derivative)

Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Unfortunately, the formula $$f = \int_{\mathbb R} \...
Riku's user avatar
  • 839
0 votes
0 answers
156 views

Function classes with high Rademacher complexity

My question is two fold, Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
gradstudent's user avatar
  • 2,246
0 votes
0 answers
76 views

Measure on infinite dimesional $L^p$ space relating size in norm to size in measure

Let $A$ be a bounded set in an infinite dimensional $L^p$ space. Fix an $\epsilon>0$. Is there a Borel measure $M$ such that $$ M(B(x,\epsilon)) \geq C, \quad \forall x \in A$$ for some $C>0$ ...
dawo's user avatar
  • 111
0 votes
0 answers
46 views

The Minkowski $(N-1)$- dimensional upper constant of a closed curve?

Let $\Omega\subset \mathbb R^N$ be open bounded smooth boundary. Let $S\subset \Omega$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<+\infty$. It is well know that if $S$ is not closed, then ...
JumpJump's user avatar
  • 679
0 votes
0 answers
104 views

Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question: Given ...
Spencer's user avatar
  • 1,771
0 votes
0 answers
148 views

existence of locally translation-invariant Borel measure on Frechet manifolds

It is well known that the only locally finite, translation-invariant Borel measure on an infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
Erik Curiel's user avatar