Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
0
votes
0answers
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.
0
votes
0answers
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 ...
-4
votes
0answers
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}{...
0
votes
0answers
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)
...
8
votes
2answers
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
votes
0answers
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
$$f(x,t_o)=\lim_{t \rightarrow t_o} \frac{\int_1^\infty g(x,t)dx}{\int_1^\infty h(x,t)dx}= ...
3
votes
1answer
100 views
A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?
Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
1
vote
0answers
44 views
Using semigroup theory for nonautonomous semilinear equations
We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...
1
vote
0answers
29 views
Are the intersection of proximinal sets in a Hilbert Space proximinal?
Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d_C$ be the infimal distance function to $C$ and $P_C(x) = C \cap S[x; d_C(x)]$ be the metric projection. We say $C$ is proximinal if $P_C(x) \...
1
vote
1answer
67 views
Density on a specific functional space.
I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let
$$
\mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{...
8
votes
0answers
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 $\...
0
votes
0answers
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?
...
2
votes
1answer
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 $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
9
votes
2answers
209 views
Explicit proof that $c_0$-module $\ell_\infty$ is not projective
It is well known in narrow circles that the homological dimension (in the sense of relative Banach homology) of $c_0$-module $\ell_\infty$ is 2. As the corollary, this module is not projective. This ...
4
votes
1answer
107 views
compact embedding for Sobolev spaces
The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$
Is it possible to determine the ...
0
votes
0answers
19 views
Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
5
votes
2answers
193 views
Making the Fourier transform quantitative
I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...
3
votes
1answer
57 views
Echange of Infimum Integral with Pointwise Infimum
Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
-2
votes
1answer
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 ...
5
votes
1answer
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 ...
5
votes
2answers
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 ...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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*?
4
votes
0answers
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 ...
2
votes
0answers
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_\...
0
votes
0answers
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 $0<s<1$.
0
votes
0answers
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 ...
0
votes
0answers
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 $...
3
votes
0answers
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 ...
1
vote
0answers
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 $\mathbb{R}^3$:
$$(Rf)(H):=\int_{l\subset ...
2
votes
0answers
44 views
Volume of critical points decreases under symmetric decreasing rearrangement
In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...
14
votes
2answers
412 views
Multiple of identity plus compact
Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
2
votes
2answers
123 views
Behaviour of Direct limit with quotient and double dual
I am trying to understand direct limit in category of $C^*$ algebras.
Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras?
Any references or ideas?
P....
3
votes
1answer
80 views
Every closed subspace $A$ of $C_0(K)$ can be regarded as a subspace of continuous functions on $A^*$?
We consider a locally compact Hausdorff space $X$ and the Banach space $C_0(X)$ of continuous functions on $X$ taking values at $\mathbb K = \mathbb R$ or $\mathbb C$, equipped with the supremum norm.
...
1
vote
2answers
146 views
A min-max approximation
Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$.
My question is : Is it true that
$$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
2
votes
1answer
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^{...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
5
votes
0answers
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 ...
0
votes
1answer
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 ...
4
votes
0answers
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 $L^{1, \delta}$?
2
votes
1answer
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 $...
4
votes
1answer
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 $...
-1
votes
0answers
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 $$\...
3
votes
1answer
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 ...
2
votes
1answer
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 $\...
14
votes
1answer
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 ...
