Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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6 views

### Definition of $C^{m,k}$-capacity of a point

I have come across the following notation and a new term $C^{m,k}$-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.

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49 views

### An exponential integral over sphere

I'm wondering if someone knows if the following kind of integral has a closed-form expression (or maybe an approximation):
$$\int_{0}^1 x^{2a} (1-x^2)^b e^{-x^2 c} dx $$
for $a,b,c > 0$. [This has ...

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56 views

### the limits of series [on hold]

let $S_n=\sum^n_{i=1} \frac{3 i^2}{4^i-1}$
$$\frac{3 i^2}{4^i-1} \leq \frac{3 i^2}{3^i}=\frac{i^2}{3^{i-1}}$$
put $A_i=\ln \frac{i^4}{3^{i-1}}$
$$A_i=4 \ln i- (i-1)\ln 3 $$
for $i>1$ $A_i=\frac{1}{...

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46 views

### Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...

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**2**answers

406 views

### Is taking the positive part of a measure a continuous operation?

Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out.
Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure ...

**-4**

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46 views

### Can we apply L'Hopital's rule with respect parameter $u$? [on hold]

If we know that $\int_1^\infty g(x,t_o)dx=0$ and $\int_1^\infty h(x,t_o)dx=0$
$ t $ is real variable here
$$f(x,t_o)=\lim_{t \rightarrow t_o} \frac{\int_1^\infty g(x,t)dx}{\int_1^\infty h(x,t)dx}= ...

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**1**answer

100 views

### A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?

Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...

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44 views

### Using semigroup theory for nonautonomous semilinear equations

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...

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29 views

### Are the intersection of proximinal sets in a Hilbert Space proximinal?

Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d_C$ be the infimal distance function to $C$ and $P_C(x) = C \cap S[x; d_C(x)]$ be the metric projection. We say $C$ is proximinal if $P_C(x) \...

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**1**answer

67 views

### Density on a specific functional space.

I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let
$$
\mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{...

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69 views

### Borel-Ecalle re-summation and resurgence: Criteria and Results

This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\...

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51 views

### $L^{\infty}-L^{\infty}$ estimates on the Schrödinger evolution

A classical estimate for solutions to Schrödinger equations are $L^1-L^{\infty}$ estimates, also known as dispersive estimates.
I wonder whether there are also $L^{\infty}-L^{\infty}$ estimates?
...

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**1**answer

137 views

### Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...

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209 views

### Explicit proof that $c_0$-module $\ell_\infty$ is not projective

It is well known in narrow circles that the homological dimension (in the sense of relative Banach homology) of $c_0$-module $\ell_\infty$ is 2. As the corollary, this module is not projective. This ...

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**1**answer

107 views

### compact embedding for Sobolev spaces

The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$
Is it possible to determine the ...

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**0**answers

19 views

### Dual representation of problems involving $f$-divergences

Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...

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**2**answers

193 views

### Making the Fourier transform quantitative

I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...

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**1**answer

57 views

### Echange of Infimum Integral with Pointwise Infimum

Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...

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**1**answer

97 views

### Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence?

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...

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**1**answer

105 views

### Non-existence of continuous extension of continuous linear operator defined on non-dense subspace

Bounded Extension from Dense Subspace Theorem. Suppose that $Μ$ is a dense subspace of a normed space $X$, that $Y$ is a Banach space, and that $T_0: Μ \to Y$ is a bounded linear operator. Then there ...

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213 views

### Bounded deformation vs bounded variation

Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...

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26 views

### A variation on Sylvester equation

Let $X$ be a finite measure space and $D,M$ be bounded linear operators on a $(L^1(X;\mathbb C))^2$. $D$ is a diagonal operator matrix whose entries are multiplication operators by the invertible ...

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40 views

### Is every nonexpansive retract of a Hilbert space closed and convex?

Given a closed and convex subset $C\subset H$ of a Hilbert space $H$, the metric projection is a nonexpansive retraction of $H$ onto $C$. This implies that every closed and convex subset of a Hilbert ...

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48 views

### Extension of a derivation on w [duplicate]

Let I be a closed left ideal of a Banach algebra A such that A has a bounded approximate identity and I just has a right approximate identity. let $D:I\to I^*$ be a derivation. Does $D$ extend to a ...

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88 views

### Minimizer of certain functional is strong $\Lambda$-minimizer

I'm interested in the quantitative isoperimetric inequality. I am currently reading https://arxiv.org/pdf/1007.3899.pdf and I have some questions regarding lemma 3.5 which states a certain situation ...

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57 views

### Extension of a derivation

Let I be a closed left ideal of a Banach algebra A and let D:I\to I* be a derivation. Does D extend to a derivation from A to A*?

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187 views

### Spectral Gap of Elliptical Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of elliptical operator $ \nabla \cdot(a(x)\nabla)$ defined on $D$, can be controlled?
The boundary condition is that the solution at ...

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76 views

### Inequality concerning BV norm

Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...

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32 views

### explicit formula for fractional laplacian.

Let $u$ be a smooth positive bounded function. Define $v(x)= \log u(x)$. Is it possible to compute $ (-\Delta)^{s} v(x)$ in terms of $u$ explicitly where $0<s<1$.

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28 views

### Relative compactness of differential operator

Let $\Omega$ be $\mathbb R^n$ or a complete (unbounded) open manifold, and $f$ be a smooth function on $\Omega$.
We consider a self-adjoint 2nd. elliptic operator $H$ on $L^2$ space(to simplify the ...

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37 views

### Density of the range of $M$ in the range of $M^{1/2}$ for $M$ positive

Let $F$ stands for a complex Hilbert space with inner product $\langle\cdot\;,\;\cdot\rangle$ and the norm $\|\cdot\|$. Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on $F$.
Let $...

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67 views

### informative examples for understanding spectral triples

I am at the beginning of my thesis work and I am trying to understand spectral triples. I can recall the definition but I have no informative examples with which to make sense of it. What are some ...

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38 views

### Kernel of Radon transform in $\mathbb{R}^3$

Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$:
$$(Rf)(H):=\int_{l\subset ...

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44 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 ...

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412 views

### Multiple of identity plus compact

Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...

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123 views

### Behaviour of Direct limit with quotient and double dual

I am trying to understand direct limit in category of $C^*$ algebras.
Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras?
Any references or ideas?
P....

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**1**answer

80 views

### Every closed subspace $A$ of $C_0(K)$ can be regarded as a subspace of continuous functions on $A^*$?

We consider a locally compact Hausdorff space $X$ and the Banach space $C_0(X)$ of continuous functions on $X$ taking values at $\mathbb K = \mathbb R$ or $\mathbb C$, equipped with the supremum norm.
...

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146 views

### A min-max approximation

Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$.
My question is : Is it true that
$$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...

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**1**answer

87 views

### Removable set for Sobolev space

It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...

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**1**answer

95 views

### Regarding approximation by invertible operators

This post here states that if $E$ is an infinite dimensional space and if $T$ is an injective, bounded,non surjective opertor with closed range in $E$, then $T$ cannot be approximated in operator norm ...

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100 views

### On Gevrey space

I want to know whether there is a dense subset of the Gevrey space $ G^s$. If not is there a standard way of constructing such dense subset?
(Eddited)The definition is as follows:
Let $s\geq 1.$ Let ...

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109 views

### Extension of elliptic complex to an exact sequence

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.
Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...

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**1**answer

103 views

### $f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?

Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...

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96 views

### An embedding question: Morrey spaces

Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?

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**1**answer

125 views

### compactness of fractional Sobolev spaces

I am looking for a reference on the paper on compact Sobolev embeddings.
If we define the Sobolev space $$X_{0}(A):=\{u\in H^s(\mathbb R^N): u=0\quad \text{in}\quad \mathbb R^N \setminus A\}$$ where $...

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**1**answer

173 views

### Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $...

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78 views

### Does this sequence have a convergence subsequence?

Let $w_n\in C([0,\tau];L^2(\Omega))$, and $\Omega$ be an open bounded set of $\mathbb{R}^2$. For every $t\in [0,\tau]$, $w_n(t)$ has a convergent subsequence in $L^2(\Omega)$. Does the sequence $$\...

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**1**answer

79 views

### Non-point spectrum for diagonalisable self-adjoint unbounded operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...

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**1**answer

63 views

### Dirac operator on manifold with periodic end

Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to
itself obtained from cutting $\...

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**1**answer

428 views

### Extreme points of convex compact sets

Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...