All Questions
10,199 questions
2
votes
1
answer
159
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
- 4,153
5
votes
0
answers
360
views
Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
- 419
1
vote
1
answer
100
views
Is there literature on the existence of solutions to elliptic systems on unbounded manifolds?
Most of the current literature I've seen is either for compact Riemannian manifolds or unbounded subsets of Euclidean space. In this article, the authors consider a priori bounds on such systems on ...
- 419
4
votes
1
answer
203
views
weights of projections and norms of operators in a von Neumann algebra
Let $M$ be an atomless von Neumann algebra equipped with a (semifinite faithful normal) weight $w$. Let $x\in M$ and let $\varepsilon>0$.
Can we find a constant $\delta>0$ such that whenever a ...
- 617
5
votes
1
answer
561
views
interiors of positive cones in ordered Banach spaces
I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references.
I want to know several examples of ...
- 79
2
votes
1
answer
184
views
Lipschitz smooth convex extension
Assume that convex $f: S \to \mathbb{R}$ with $L$-Lipschitz continuous gradient on some convex compact $S \subset \mathbb{R}^d$ is given. It would be very helpful if there existed function $F$ such ...
1
vote
1
answer
90
views
The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
- 4,058
8
votes
0
answers
192
views
Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
- 60.5k
1
vote
1
answer
100
views
Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?
Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let
$f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
- 13
1
vote
1
answer
133
views
Smoothness of an equivalent norm
For an arbitrary set $\Gamma$, Day's norm on $c_0(\Gamma)$ is defined by
$$ \Vert x \Vert = \sup \bigg \{ \bigg ( \sum_{k=1}^n 4^{-k} x^2(\gamma_k) \bigg )^{\frac{1}{2}} : (\gamma_1, \cdots, \gamma_n) ...
- 85
7
votes
2
answers
320
views
Uniqueness of left-invariant Borel probability measure on compact groups
On a compact topological group, consider two left-invariant probability measures $\mu$ and $\nu$ defined on the Borel sigma-algebra. Is it true that they coincide?
It is classical that the Haar ...
2
votes
1
answer
264
views
Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?
As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
- 1,179
3
votes
2
answers
294
views
Domain of spectral fractional Laplacian
Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
- 1,171
1
vote
1
answer
113
views
The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?
Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as
\begin{equation}
P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n
\end{equation}
...
- 3,477
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
5
votes
1
answer
221
views
Arens regularity of $\mathrm{BV}(\mathbb{R})$
$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of ...
- 6,406
4
votes
0
answers
242
views
On the Dunford-Pettis property and multiplier algebras
I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that:
Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
2
votes
1
answer
623
views
On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
- 101
2
votes
0
answers
62
views
Localized estimate for divergence free vector field
Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
- 31
1
vote
1
answer
256
views
Moser iteration in dimension $6$
Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying
\begin{align*}
\Delta f \leq gf-\frac{3}{4}f^2
\end{align*}
Where $g$ is another smooth ...
- 954
1
vote
1
answer
69
views
Local completion of bornological space
I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space.
The construction is as follows:
A Hausdorff locally convex space $E$ is locally ...
- 651
5
votes
1
answer
512
views
Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
- 75
0
votes
0
answers
119
views
About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
5
votes
1
answer
615
views
Is every character of the algebra of continuous functions on a locally compact space some evaluation?
Given any locally compact Hausdorff space $X$, let $C(X)$ denote the complex algebra of all complex-valued continuous functions on $X$.
Question. Given an arbitrary character (i.e. a non-zero ...
- 960
4
votes
1
answer
291
views
Structure of all Wightman QFTs
I have two related questions related to constructive/axiomatic QFT.
Is there a structure on the collection of all QFTs, as defined by the Wightman axioms? Do they form some type of category?
...
- 43
0
votes
0
answers
155
views
Reducing subspaces of unitary operators
Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
- 23
1
vote
1
answer
120
views
Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
- 227
1
vote
0
answers
119
views
Continuity of a minimizing measure w.r.t a parameter
Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
- 41
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
1
vote
1
answer
159
views
Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?
Let $X$ be an operator space and $CB(X)$ be the set of all completely bounded linear maps $f: X \to X$. Note that $CB(X)$ becomes a Banach algebra for the composition of operators.
Is the ...
- 175
5
votes
1
answer
542
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
- 1,075
1
vote
0
answers
178
views
Trans-universality for finite-dimensional Banach space
In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case:
QUESTION: does there exist a ...
- 4,786
3
votes
2
answers
429
views
Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 4,074
1
vote
1
answer
115
views
What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?
$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals ...
- 515
2
votes
1
answer
165
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
- 1,043
2
votes
0
answers
320
views
Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
2
votes
0
answers
329
views
Conditions for an existence of smooth solution to a parabolic PDE
I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset):
\begin{equation*}
u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
- 425
2
votes
0
answers
157
views
Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
3
votes
1
answer
198
views
Do radially bounded sets form a bornology?
We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
- 5,529
6
votes
0
answers
253
views
Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
- 25.8k
2
votes
1
answer
150
views
Sufficient conditions for the graph measurability of a multivalued function
I am currently working on a problem related to the measurability of multi-functions in the context of mathematical economics. Specifically, I am searching for sufficient conditions regarding the graph ...
- 79
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
2
votes
1
answer
225
views
Boundary points in $\overline{\operatorname{conv}\{z_i\}_{i\in I}}$
Let $X$ be an infinitely-dimensional Banach space and $\{z_i\}_{i\in I}$ be a set of linearly independent points in $X_{\leq 1}$, the closed unit ball of $X$. $I$ the index set is not necessarily ...
- 307
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 434
1
vote
1
answer
143
views
$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
- 101
1
vote
0
answers
54
views
Minimal F-semi-norms
There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...
- 5,529
1
vote
2
answers
181
views
Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$
Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE:
$$\Delta f -\frac{1}{2}h f = 0$$
where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
- 3,535
1
vote
1
answer
310
views
Weak convergence in $H^{1}$ implies different convergence in $L^{p}$?
Suppose I have a sequence $\{f_{n}\}_{n\in \mathbb{N}} \subset H^{1}(\mathbb{R}^{d})$ which converges weakly to $f$ in $H^{1}(\mathbb{R}^{d})$, in the sense that $\langle f_{n},\varphi \rangle_{L^{2}}+...
1
vote
1
answer
84
views
optimization over moving domains
Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem:
$$L(a)=\inf_{b\in B_a}\ell(b),$$
where $\ell(b)$ is a infinite-times ...
- 87
3
votes
1
answer
339
views
On a Poincaré inequality with weight
Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents.
Is it true that there exists a ...
- 1,263