**1**

vote

**1**answer

96 views

### Are most random variables trivially sub-gaussian? [on hold]

I'm trying to understand sub-gaussian RVs to see if they could be relevant to my work.
The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform ...

**3**

votes

**1**answer

77 views

### A relative property gamma and $L(\mathbb F_2)$

Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?

**2**

votes

**1**answer

118 views

### Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem.
Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...

**16**

votes

**0**answers

339 views

+500

### Separating pure states on the $2\times 2$ matrix algebra

I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C*-algebra of ...

**2**

votes

**0**answers

123 views

### Why only Normed Linear Spaces? [on hold]

It is well known that "Norm on a vector space can be used to obtain a metric on that space."
I think easily we can generalize the notion of norms to groups and rings.
My questions are,
Why ...

**4**

votes

**0**answers

88 views

### Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...

**10**

votes

**2**answers

887 views

### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

**4**

votes

**0**answers

67 views

### Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...

**1**

vote

**0**answers

35 views

### Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...

**0**

votes

**0**answers

14 views

### c-superdifferential is unique +cost function is differentiable, then the potential function is differentiable?

Let $M$ be a compact Riemannian manifold, $\mu$ and $\nu$ are two Borel probability measures, the cost function $c(x,y)=\frac{d^2(x,y)}{2}$.
It's well known that the infimum of the Kontorovich's ...

**0**

votes

**1**answer

63 views

### Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...

**10**

votes

**1**answer

896 views

### Matrices with entries in a $C^*$-algebra

Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...

**4**

votes

**0**answers

97 views

### Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set ...

**1**

vote

**0**answers

100 views

### Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...

**0**

votes

**0**answers

88 views

### A question about tensor product [on hold]

For every $f, g$ in $L^1(G)$, we know the function $[(x,y)↦f(x)g(y)]$ belongs to $L^1(G\times G)$ where here $\times$ means cartesian product. Why are these functions dense in $L^1(G\times G)$?

**5**

votes

**1**answer

305 views

### Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and
$\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$
is finite.
There are at least two ...

**3**

votes

**1**answer

103 views

### Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?

**6**

votes

**0**answers

177 views

### Does hypoellipticity imply the existence of a parametrix?

Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...

**0**

votes

**0**answers

50 views

### Norm inequalities between difference operators

Assume $(v_i)$ to be a sequence in $\ell_\infty(\mathbb{R})$ for $i=1,\dotsc,N$. Define the difference operator as $\Delta v_i:=v_{i+1}-v_i$ and $\Delta^n v_i:=\Delta(\Delta^{n-1} v_i)$. Then, how can ...

**1**

vote

**0**answers

86 views

### One-sided local $L^p$ spaces

Consider the vector space $L^p_{\text{left-loc}}$ of measurable functions $f:[0,1]\to\mathbb R$ so that for all $x\in(0,1]$ there exists $\delta>0$ so that $f|_{[x-\delta,x]}\in L^p$.
Does this ...

**6**

votes

**2**answers

231 views

### Non strictly-singular operators and complemented subspaces

If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed and complemented, subspace $Y$ such that $T$ ...

**5**

votes

**1**answer

186 views

### Hahn Banach type extension of a Lipschitz map

The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by ...

**0**

votes

**1**answer

121 views

### Continuous maps on compact topological spaces which induce compact (Fredholm) operators

Let $X$ be a compact Hausdorff space. A continuous map $f:X \to X$ defines a bounded linear operator $T_{f}$ on the Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi\; \text{is continuous}\}$ with ...

**2**

votes

**1**answer

104 views

### Conformally covariant distributions

In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form
$$
\langle O(x)O(y)\rangle,
$$
where $O$ is a scalar primary field. Scale covariance ...

**1**

vote

**0**answers

170 views

### system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [closed]

What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$?
Define $F$ in the domain where a continuous function exists that behaves so for ...

**3**

votes

**1**answer

310 views

### Survey papers on the role played by PDE in mathematics

There are already several questions on Mathoverflow about the application of PDE to several other topics (e.g., algebraic and differential geometry and topology, number theory, harmonic analysis, ...

**2**

votes

**1**answer

252 views

### The Tensor product of algebra group

Let G is a locally compact group. Is the following true?
The tensor product of $L^1(G)$ with $L^1(G)$ is $L^1(G \times G)$.

**3**

votes

**1**answer

71 views

### Existence of a countable linear combination with positive coefficients

Consider a (Hausdorff and complete) locally convex topological vector space $V$ and a countable subset $(v_k)_{k=1}^\infty \subset V$ of non-zero vectors.
$(*)$ Under what conditions on this ...

**5**

votes

**2**answers

226 views

### Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...

**1**

vote

**0**answers

34 views

### The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.
Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...

**1**

vote

**0**answers

81 views

### Regularity of weak solution

I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...

**4**

votes

**2**answers

312 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**5**

votes

**1**answer

394 views

### A problem in functional calculus

This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very ...

**3**

votes

**1**answer

89 views

### Generalized functions on a product of two manifolds

Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map ...

**0**

votes

**0**answers

142 views

### Is possible a version of abc conjecture for operators, it is in the context of functional analysis? [closed]

It is well known the following unsolved problem,
abc conjecture. For an integer $n\geq 1$, we define the radical of $n$, as the product of the distinct prime factors of n, $rad(n)=\prod_{p|n}$ with ...

**-1**

votes

**0**answers

150 views

### Can test Riemann Hypothesis by interpolation Banach couples? [closed]

In [1], a paper deals to Riemann Hypothesis, we can read the following statment
The Nyman-Beurling Riemann Hypothesis holds if and inly if
$$span_{L^{2}(0,1)}\{\eta_{\alpha}, ...

**1**

vote

**0**answers

46 views

### “Friedrichs extension Laplacian” vs “Weak Laplacian” and fractional powers

Take $\Omega$ to be a bounded domain and consider Neumann BCs.
In some works, I see that a Laplacian $(-\Delta_D)^{\frac 12}$ is defined as an operator with domain $H^1(\Omega)$, and in other works, ...

**0**

votes

**1**answer

88 views

### solution uniqueness of non-linear Fredholm equations

the equation is
$F(x)=G(\int k(x,y)f(y)dy)$ $(*)$
where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...

**1**

vote

**1**answer

280 views

### Where to learn about parabolic Hölder spaces and when to use them

Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...

**2**

votes

**1**answer

70 views

### Associative convolution on p-adic distribution

Let $\mathcal{D}(\mathbb{Q}_p)$ be the space of the locally constant functions with compact support and let $\mathcal{D}'(\mathbb{Q}_p)$ be the space of distributions: linear functionals on ...

**1**

vote

**0**answers

50 views

### Fractional Sobolev space as a Cameron-Martin space

In their paper "On Fractional Brownian Processes" (link here to the working paper), Feyel and Pradelle (1997) write in the introduction: "we give very simple proofs of the existence of different ...

**5**

votes

**3**answers

574 views

### Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by
$$
y''+(\delta(x)-\lambda^2)y=0.
$$
Then, to find ''bound states'', you solve on the right and find the ...

**4**

votes

**2**answers

794 views

### Heat kernel estimates and Gaussian estimates for semigroups, good reference?

Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.
If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ ...

**0**

votes

**0**answers

135 views

### How to change the given metric if we want to add few extra isometries?

I have a Hilbert Space $X$ and a group $G$, which consists of bounded linear self-bijections of $X$ (if it helps, this group has a locally compact, but not compact topology). Is there a canonical way ...

**6**

votes

**2**answers

293 views

### Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and
g is radially symmetric,
the function $ (0, \infty )\ni t \mapsto g ...

**1**

vote

**2**answers

168 views

### Derivative of Band-limited functions [closed]

I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy
$$\int_{-\infty}^\infty f(x)^2dx=c.$$
(For pure mathematicians: "bandlimited" means ...

**3**

votes

**1**answer

211 views

### reference request: direct product of WOT-continuous unitary representations

In an article I'm revising, I spend some time giving a self-contained proof of the following result
Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...

**0**

votes

**0**answers

67 views

### Cuntz comparison of strictly positive elements in finite C*-algebras

Let $A$ be a finite, non-unital C*-algebra, $s\in A$ a strictly positive element and $a\in A$ a positive element that is Cuntz-equivalent to $s$, i.e. there exist sequences $\{x_n\},\{y_n\}\subset A$ ...

**2**

votes

**2**answers

131 views

### Composition operators on fractional-order (periodic) Sobolev spaces

(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...

**1**

vote

**1**answer

66 views

### question about $TGV^2$ space

Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and
$$
TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in ...