**14**

votes

**9**answers

10k views

### A book for problems in Functional Analysis

I want to know if there's any book that categorizes problems by subjects of Functional Analysis.
I'm studying Functional Analysis now a days and I really need to solve some problems in order to ...

**2**

votes

**1**answer

84 views

### understanding of rough path

A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...

**5**

votes

**1**answer

58 views

### Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...

**4**

votes

**0**answers

44 views

### Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ...

**4**

votes

**0**answers

40 views

### Specific type of Carleman Estimate

Suppose that in a compact Riemannian manifold with boundary one has the following type of carleman estimate:
$$ \| e^{\tau \phi} \triangle_g e^{-\tau \phi} u\|_{L^2(M)}\ge C \tau \|u\|_{L^2(M)} $$
...

**2**

votes

**0**answers

28 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 $\mathcal{D}'$ be the space of distributions: linear functionals on $\mathcal{D}$. In the ...

**1**

vote

**0**answers

12 views

### Counting variables to look for invariances/range conditions

A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...

**2**

votes

**1**answer

133 views

### A question on the Frechet derivative

Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...

**-4**

votes

**0**answers

71 views

### Is it possible to compare Sobolev space and Polish space? [on hold]

This question was asked in math.stackexchange.com
http://math.stackexchange.com/questions/1274873/is-it-possible-to-compare-sobolev-space-and-polish-space
I did not get any comment or reply so I am ...

**0**

votes

**0**answers

25 views

### Preimage of singular points of smooth map between vector space and $SU(n)$

(Moved from Math SE as no answer was forthcomming: http://math.stackexchange.com/q/1294521/161684)
Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ (which is taken to be surjective) ...

**0**

votes

**1**answer

35 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 ...

**0**

votes

**1**answer

74 views

### Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all:
Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...

**3**

votes

**2**answers

145 views

### $\mathcal S'(\mathbb R^d)$ is separable [on hold]

I Think the statement is true, but I struggle to find a reference for the fact that the space of tempered distributions equipped with the weak-* topology is separable.
Thank you for your help!

**0**

votes

**0**answers

32 views

### interpolation between Bochner spaces

Is there a reference for the interpolation result stating the existence of an embedding
\begin{equation}
L^2(I;W^{2,p}(\Omega)) \cap H^1(I;L^p(\Omega)) \hookrightarrow ...

**2**

votes

**0**answers

53 views

### Sobolev embedding on warped product

Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...

**2**

votes

**1**answer

214 views

### Complete solution set of a Convolutional Equation?

Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..
Setup: In what ...

**2**

votes

**0**answers

42 views

### Multiplication operators are sectorial [migrated]

Consider the multiplication operator $M_a$ on $L^p(M)$, where $M$ is a Riemannian manifold, and $a$ is a non-negative function. An operator $A$ is said to be sectorial if there exists $\theta \in (0, ...

**6**

votes

**2**answers

203 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

**0**answers

49 views

### Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that ...

**2**

votes

**0**answers

62 views

### Resolvent estimate of hyperbolic Laplacian [on hold]

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form
$$\Vert (-\Delta - \lambda ...

**0**

votes

**0**answers

44 views

### Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem
that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$,
then ...

**2**

votes

**1**answer

218 views

### Interpolation between weighted $L^p$ spaces

Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded.
Let $T$ be the ...

**1**

vote

**1**answer

83 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**0**

votes

**0**answers

43 views

### Alternating projections and trace preserving maps

Let $\{\Pi_i\}_{i=1}^N$, $\Pi_i\in \mathbb{C}^{n\times n}$ be a set of orthogonal projections. By the Von Neumann alternating projection theorem, it holds that
$$
...

**6**

votes

**1**answer

272 views

### An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...

**0**

votes

**0**answers

57 views

### Multiple convex sets hyperplane separations

Hyperplane separation theorem states that if there is a bounded convex set $X$, a convex set $Y$, then there is a hyperplane that separates $X$ from $Y$.
That is, there is a statement that gives ...

**2**

votes

**0**answers

32 views

### Smooth bivariate functions identifiable under permutations

Consider a "smooth" $f : [0,1]^2 \to [0,1]$ and let $\pi : [n] \to [n]$ be a permutation of $[n] =\{1,\dots,n\}$. Let us define
$$
A^{f,\pi} := \Big( f\Big(\frac{\pi(i)}{n},\frac{\pi(j)}{n}\Big) ...

**1**

vote

**0**answers

87 views

### Transformation of kernel

I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if ...

**3**

votes

**2**answers

161 views

### Examples of TVS with no non-trivial open convex subsets

I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...

**2**

votes

**1**answer

150 views

### Contraction semigroup

Let $(X, \mu)$ be a finite measure space and let $A$ be a non-negative self-adjoint operator which generates a contraction semigroup $e^{tA}$ on $L^2(X, \mu)$. If additionally, we have that $e^{tA}$ ...

**0**

votes

**0**answers

30 views

### Powers of compact operators [migrated]

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...

**5**

votes

**2**answers

228 views

### Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side:
$$
\Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial ...

**3**

votes

**0**answers

117 views

### Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?

Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in ...

**3**

votes

**0**answers

66 views

### independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper:
Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...

**2**

votes

**1**answer

606 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**5**

votes

**1**answer

95 views

### Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...

**3**

votes

**1**answer

176 views

### Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples:
$\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
$\sum_{-\infty}^{\infty} ...

**6**

votes

**2**answers

316 views

### Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...

**1**

vote

**0**answers

69 views

### Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?)
But I would like to know if ...

**9**

votes

**1**answer

240 views

### Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...

**1**

vote

**1**answer

93 views

### Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...

**0**

votes

**0**answers

62 views

### Existing complete function space under suitable norm

This question was asked in math.stackexchange.com but no suitable answer was received, so I am posting it here.
This is a question which came to me due to several previous question: sorry for the all ...

**2**

votes

**1**answer

69 views

### Integral kernels of self-adjoint operators [closed]

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...

**0**

votes

**0**answers

66 views

### Sobolev trace theorem

Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$,
where $\Omega$ is knows as a bounded domain
with smooth boundary $\partial D$.
We choose any subdomain $D\subset Q$
with smooth boundary $\partial ...

**0**

votes

**0**answers

35 views

### Bounded linear functionals over smooth maps of a compact interval [migrated]

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...

**3**

votes

**2**answers

191 views

### When does a function space allow for point evaluations? [closed]

Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology.
What are necessary and sufficient conditions for point evaluations at arbitrary $x ...

**4**

votes

**1**answer

96 views

### Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$

Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$
with $f_{ji}:H_i \to H_j$ being the trace class ...

**2**

votes

**1**answer

164 views

### Multivariate Maximal Hilbert Transform

One way to define the maximal Hilbert transform of a function, $f$, is by
$$\mathcal{H}[f](x):=\underset{\varepsilon>0}{\sup}\;\left|\int_{|x-t|\geq\varepsilon}\dfrac{f(t)}{x-t}dt\right|,\quad ...

**3**

votes

**1**answer

97 views

### Can you pair $H^s(\Omega)$ and $H^{-s}(\Omega)$ on a domain $\Omega$?

Consider the fractional Sobolev spaces on $\mathbb R^n$
$H^s(\mathbb R^n) := \left\{ u \in \mathcal S'(\mathbb R^n) \; : \; ( 1 + |\xi|^2 )^{s/2} \hat u \in L^2(\mathbb R^n) \right\}$.
Let $\Omega$ ...

**3**

votes

**1**answer

111 views

### Asymptotics of Fresnel integrals

It is known that
\begin{equation*}
I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x
\end{equation*}
is a bounded ...