All Questions
1,150 questions
4
votes
1
answer
365
views
Lusin Lipschitz approximation in BV and Sobolev space
Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that
Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
- 839
4
votes
0
answers
148
views
Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
- 91
4
votes
2
answers
378
views
Basic properties of expectation in non-separable Banach spaces
$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$
Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear ...
- 931
4
votes
1
answer
879
views
Commuting with an unbounded operator
Let $H$ be a Hilbert space. Let $A$ be a closed unbounded operator, and let $B\in B(H)$ be a bounded operator.
Definition:
$A$ and $B$ strong-commute if the partial isometry in the polar ...
- 43.2k
4
votes
0
answers
282
views
Pitfalls when generalizing the heat kernel of a Riemannian metric
Suppose $M$ is a Riemannian manifold with some compact quotient under isometries.
Associated with the Riemannian metric one has the Laplace-Beltrami operator $\Delta$ and the heat kernel $p(t,x,y)$ ...
- 4,304
4
votes
1
answer
279
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
4
votes
1
answer
222
views
A continuous bi-Lipschitz shrinking of a domain into a compact subset
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is:
(1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...
- 401
4
votes
2
answers
367
views
Fokker-Planck equation for a truncated process
Let $X_t = x + bt + \sigma W_t$ be an arithmetic Brownian motion, where $x$ is a random variable independent to $W$, and $\sigma>0$. Suppose the initial distribution is given by $\mathbb P(X_0 \in ...
- 1,399
4
votes
0
answers
115
views
Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$
I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly.
Question 1. In the ...
- 131
4
votes
1
answer
414
views
Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight
The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form
$$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
- 324
4
votes
0
answers
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
4
votes
2
answers
728
views
How to find the symmetry group of a differential equation
If one is given a differential equation, e. g. the KdV equation $\ u_t + u_{xxx} + uu_x = 0$, how can he find all of the symmetries of the differential equation? Is there also a method that works for ...
- 3,738
4
votes
2
answers
410
views
Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$
The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function).
$$(x^2y')'-x^2y=\lambda \;y$$
Now for a higher-degree ...
- 1,199
4
votes
2
answers
883
views
Principal symbol for non-linear differential operators
$\newcommand{\R}{\mathrm{R}} \newcommand{\N}{\mathrm{N}}\newcommand{\DD}{\mathrm{D}}\newcommand{\dd}{\mathrm{d}}$
Prerequisites: Let $\mathrm{T}: C^\infty(\Omega) \rightarrow
C^\infty(\Omega), u(\...
- 381
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284
4
votes
2
answers
434
views
A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
- 433
4
votes
3
answers
522
views
Can we specify the value of harmonic forms at a point?
Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed.
Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$.
Does there exist an open ...
- 6,741
4
votes
1
answer
412
views
Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
- 617
4
votes
1
answer
964
views
Convergence of Fredholm determinants
Let $(X_N)_N$ be a sequence of trace class operators acting on, say, $L^2(\mathbb{R})$. What are the minimal assumptions in order to have the convergence of their Fredholm determinant
$$
\lim_N\det(...
- 2,135
4
votes
1
answer
311
views
Examples of Borel probability measures on the Schwartz function space?
Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...
- 3,477
4
votes
1
answer
473
views
Is the set of weak*-continuous operators closed in the weak*-operator topology?
I recently came across this unanswered MO question an answer to which I would also be interested in. However the formulation of said question is somewhat imprecise and lacking detail in my opinion so ...
4
votes
1
answer
86
views
Approximation of multipliers by multipliers of a smaller set
Let $X$ be a compact metric space, and let $B$ be a convex balanced bounded set in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$.
Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B\...
- 5,529
4
votes
5
answers
3k
views
Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 5,327
4
votes
1
answer
282
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
4
votes
2
answers
257
views
A functional inequality which calculates the limitation of human eyes
Find all pair of function $f^-,f^+:[0,1]\rightarrow[0,1]$ such that:
(1)$f^-(x)\leq x\leq f^+(x)$.
(2)$f^-(x)+f^+(1-x)=1$.
(3)$f^-(x)f^-(y)\leq f^-(xy)\leq f^-(x)f^+(y)$.
(4)$f^+(x)f^-(y)\leq f^+(xy)\...
- 1,462
4
votes
1
answer
379
views
A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
4
votes
1
answer
520
views
Compactly generated Banach spaces
Suppose that $X$ is a Banach space (or more generally, Frechet space) such that $X$ is the closure of the span of a compact (in the original topology) subset $K$. Do we know anything "nice" about $X$, ...
- 3,115
4
votes
0
answers
126
views
Relationship between three different definitions of solutions for ODE with irregular coefficient
What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
- 839
4
votes
0
answers
281
views
Dual space of ${\rm Lip}_0(\mathbb R^d)$
This question comes to me when I read this paper : https://arxiv.org/pdf/1702.06049.pdf
Let ${\rm Lip}_0(\mathbb R^d)$ be the space of Lipschitz functions $F$ on $\mathbb R^d$ with $F(0)=0$. Then is $...
user128095
4
votes
1
answer
1k
views
Existence of solutions to first-order PDE involving convolution
Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation,
\begin{align}
\frac{\partial}{\partial \alpha} ...
- 193
4
votes
0
answers
310
views
Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$
For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...
- 6,853
4
votes
1
answer
228
views
Haar-null union of dense subsets
Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...
- 63
4
votes
2
answers
374
views
Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
4
votes
0
answers
75
views
The sum of linear partial differential operators of equal strength
If $P$ and $P'$ are linear partial differential operators with constant complex coefficients on $U = \mathring U \subseteq \Bbb R^m$, we say that $P \sim P'$ if and only if $\dfrac {\tilde P} {\tilde {...
- 5,407
4
votes
1
answer
174
views
A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,529
4
votes
1
answer
394
views
Linear PDE, analytic continuation, Green's function and boundary conditions
I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a ...
- 1,150
3
votes
1
answer
250
views
Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
- 1,835
3
votes
0
answers
173
views
$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature
This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ .
The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
- 155
3
votes
4
answers
2k
views
Suitable references for the the Stone-von Neumann Theorem
Hi all,
I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of ...
- 1,719
3
votes
0
answers
411
views
Bounded functions dense in Sobolev Spaces
Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
- 9,853
3
votes
2
answers
280
views
Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421
3
votes
0
answers
87
views
Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
3
votes
1
answer
329
views
Survey on functional equations and inequalities
Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
user60665
3
votes
0
answers
443
views
Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?
My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call $...
- 1,199
3
votes
0
answers
144
views
Transitive closure of balanced bounded mass transport
Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
- 19.7k
3
votes
1
answer
1k
views
A calculus question related to the nonnegative definite functions
I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
- 1,649
3
votes
1
answer
233
views
A special approximation of the Heaviside function
Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function such that
$$f_\epsilon(x) = f_1(x/\epsilon) = \begin{cases} 0 & \text{ if } x < 0 \\
1 & \text{ if } x/\epsilon \ge 1
\...
- 131
3
votes
2
answers
470
views
If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer ...
- 1,396
3
votes
1
answer
383
views
"Nice" functions on infinite-dimensional space of germs of continuous functions at a point
Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
- 24.9k
3
votes
1
answer
368
views
Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?
Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...