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Non-linear weak*-continuous left inverses

Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...
T. Milva's user avatar
3 votes
1 answer
1k views

"Relative compactness of a family of probability measures" and relative compactness & sequential compactness of sets

I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable. In the discussion ...
Hyeon Lee's user avatar
3 votes
0 answers
217 views

Hardy Littlewood maximal function bounds

Let $u \in W^{1,p}(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following maximal function $$ ...
Adi's user avatar
  • 455
3 votes
0 answers
102 views

Determining what happens to the spectrum of Schrödinger operator as boundary condition changes

I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck. Suppose I have a metric graph $G$ (or even a closed interval, to make ...
GSofer's user avatar
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0 answers
87 views

Doubt when calculating the S-transform of Hida differential operator

Assume we have a Hida test function $\varphi\in (\mathcal S)$, and $y\in \mathcal S'(\mathbb R)$. Define the Gateaux directional derivative of $\varphi$ (in the direction of $y$) by: $$D_y\varphi(x):=\...
Chaos's user avatar
  • 515
3 votes
0 answers
382 views

Green's function for Robin boundary condition

Let $\Omega$ be a bounded subset in $\mathbb{R}^n$, $n \ge 3$, with smooth boundary $\partial \Omega$. Assume given $a^{ij} \in W^{2, p}(\Omega)$ with $a^{ij} = a^{ji}$, $f \in L^p(\Omega)$ and $g \in ...
Romain Gicquaud's user avatar
3 votes
0 answers
127 views

Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?

My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider, $\dot{x}=Ax$, where $x$ is the infinite dimensional ...
Piyush Grover's user avatar
3 votes
0 answers
251 views

Eigenvalue bounds and triple (and quadruple, etc.) products

Very basic and somewhat open-ended question: Let $A$ be a symmetric operator on functions $f:X\to \mathbb{R}$, where $X$ is a finite set. Assume that the $L^2\to L^2$ norm of $A$ is $\leq \epsilon$, i....
H A Helfgott's user avatar
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3 votes
0 answers
73 views

A holomorphic shrinking of a domain into a compact subset

This question is related to these two. Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$...
erz's user avatar
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Is there a complex Katetov-Tong theorem?

For any bounded function $f: X \to \mathbb R$, not-necessarilly continuous, one can define for any $x$, the real functions $$ \limsup f(x) = \inf_{U\in\mathcal V_x} \sup_{u\in U} f(u) $$ and $$ \...
André Porto's user avatar
3 votes
0 answers
113 views

Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe $$ \tilde f = f \star h + n $$ where $\star$ is convolution, $h$ is the point spread function (caused,...
Elena Yudovina's user avatar
3 votes
0 answers
89 views

Error rate implying regularity

My question is a bit general/vague. It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...
user69642's user avatar
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186 views

How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
Tao's user avatar
  • 429
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0 answers
74 views

Equivalence relation induced by Kolmogorov quotients

Recall: given a (possibly non-$T_0$) topological space $X$, its Kolmogorov quotient $KX$ is the $T_0$ topological space formed by $X/\sim$ where $x\sim y$ if they are topologically indistinguishable. ...
Willie Wong's user avatar
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3 votes
0 answers
232 views

Does the suspension spectrum functor preserve weak equivalences?

Let $\Sigma^{\infty}$ denote the suspension spectrum functor from pointed topological spaces (=CGWH spaces) to orthogonal spectra. As usual, a weak equivalence of spaces is a continuous map inducing a ...
nikola karabatic's user avatar
3 votes
0 answers
26 views

Does a compact ANR have a local equiconnecting function which connects distinct points by simple paths?

It is known that if $X$ is a (metric) ANR, then $X$ is locally equiconnected, that is, there is a neighborhood $V$ of the diagonal $\Delta X \subseteq X \times X$ and a continuous function $$f \colon ...
Cihan's user avatar
  • 1,726
3 votes
0 answers
393 views

On a possible attempt to prove the invariant subspace problem

This question involves a possible method to prove the invariant subspace problem for (separable) infinite dimensional Hilbert spaces. The idea comes from various results on this topic; more precisely, ...
Manuel Norman's user avatar
3 votes
0 answers
68 views

A strange convergence for a semigroup of operators

I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows: Let $A,B$ ...
Alex M.'s user avatar
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0 answers
130 views

Question about a paper on approximate identities

I am currently reading this paper on approximate identities of ternary Banach algebras. Assume that $(A, [.,.,.])$ is a ternary Banach algebra. A bounded net $(e_{\alpha}, f_{\alpha})$ is said to be ...
Math Lover's user avatar
  • 1,115
3 votes
0 answers
342 views

A.C.M. van Rooij's *Non-archimedean functional analysis* (1978) is very out-of-print! Anyone know of any good alternatives?

(This is a literature/reference question.) So... long story short: (1) In my present research, I needed a theory of continuous functions from the $p$-adic integers to the $q$-adic integers. Unable ...
MCS's user avatar
  • 1,284
3 votes
0 answers
91 views

How does one define the gradient of a Markov semigroup?

In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper: ...
0xbadf00d's user avatar
  • 167
3 votes
0 answers
143 views

Does the compact-open topology retain topological groups?

Let $X$ be a topological space and $Y$ a topological group. Then $C(X,Y)$ is a group, and can also be endowed with the compact-open topology. Is $C(X,Y)$ in the compact-open topology necessarily a ...
Eli Falk's user avatar
3 votes
0 answers
165 views

Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
Daron's user avatar
  • 1,955
3 votes
0 answers
222 views

Sets of finite perimeter: intersection with an half space

I have a question regarding sets of finite perimeter. In particular I'm interested to find $$\mu_{E \cap H_t}, \label{1}\tag{1}$$ where $E$ is a set of finite perimeter in a generic open set $\Omega \...
ty88's user avatar
  • 51
3 votes
0 answers
82 views

Compatibility between the source and the boundary condition for an Helmholtz-type equation

Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
Capublanca's user avatar
3 votes
0 answers
85 views

A spectral characterization of path connected spaces

Let $X$ be a compact Haussdorf topological space with the following property: For every continuous function $f:X\to \mathbb{C}$ the image $f(X)$ is a path connected subset of $\mathbb{C}$. Is such ...
Ali Taghavi's user avatar
3 votes
0 answers
269 views

Kazhdan Property T of semisimple Lie groups

I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M., Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259] (MSN). I want to ...
A beginner mathmatician's user avatar
3 votes
0 answers
58 views

Criteria for density of subgroup of diffeomorphism group

Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
ABIM's user avatar
  • 5,405
3 votes
0 answers
192 views

Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform. ...
LL 3.14's user avatar
  • 230
3 votes
0 answers
188 views

Invariant subspaces of Markov operators

I am currently working on some kind of graph theoretic problem and the following question came up: Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...
Yaddle's user avatar
  • 381
3 votes
0 answers
117 views

Optimal Poincaré constants under combined boundary and average conditions

Let $\Omega=[0,1]^2$ be the unit square, $\Gamma_1=[0,1]\times\{0;1\}$ its horizontal boundary and $\Gamma_2= \{0;1\}\times[0,1]$ its vertical boundary. I would like to know the optimal Poincaré ...
DiegoG7's user avatar
  • 53
3 votes
0 answers
134 views

Colimits of weak Hausdorff $k$-spaces

Notations: $\mathbf{T}$ is the category of weak Hausdorf $k$-spaces. $\mathbf{K}$ is the category of $k$-spaces. Fact: The inclusion functor $\mathbf{T} \subset \mathbf{K}$ is a right adjoint. It ...
Philippe Gaucher's user avatar
3 votes
0 answers
148 views

Markov semigroups and resolvents, difference of continuity

Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define \begin{align*} ...
sharpe's user avatar
  • 721
3 votes
0 answers
164 views

On Pitt's inequality (weighted Fourier inequality)

One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
DSM's user avatar
  • 1,216
3 votes
0 answers
61 views

Boundedness of $\chi_{\{f_n=0\}}$ in the BV norm

Let $f_n \in H^2(\Omega) \cap C^0(\bar \Omega)$ be a sequence of functions that are uniformly bounded in $H^2(\Omega) \cap C^0(\bar \Omega)$ on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ...
FavorExistingPopularTags's user avatar
3 votes
0 answers
170 views

singular support in the singular case

For any constructible sheaf (or D-module) $\mathcal{F}$ over a smooth variety $X$ over $\mathbb{C}$, there is a notion of singular support $SS(\mathcal{F})$ that lives in the cotangent bundle $T^{*}X$ ...
prochet's user avatar
  • 3,472
3 votes
0 answers
63 views

Continuity of local spectral radius

Let $H$ be a complex Hilbert space and let $T \in \mathcal{B}(H)$ be a linear, bounded operator. Given $x \in H$ we define its local spectral radius as $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|...
javi1996's user avatar
  • 355
3 votes
0 answers
109 views

Integral kernel of composition of linear operators acting in $L^2(\mathbb{R})$

Let $A$ and $B$ be linear operators acting on $L^2(\mathbb{R})$ or some dense subset of that space. We assume that they are integral operators (possibly with distributional kernel) $Af(x)=\int_{\...
Adam's user avatar
  • 31
3 votes
0 answers
646 views

On properties on a certain functional

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...
bambi's user avatar
  • 375
3 votes
0 answers
89 views

Reference request: Projection operators in metric spaces

Given a metric space $(X,d)$ and a subset $S\subset X$, the projection $P_S$ onto $S$ is well-defined as a set valued function. I am interested in learning more about properties of these projections ...
JohnA's user avatar
  • 710
3 votes
0 answers
84 views

Norm-controlled inverses vs uniform openness of multiplication

Let $A$ be a unital commutative Banach algebra and let $\hat{a}\in C(\Phi_A)$ be the Gelfand transform of an element $a\in A$. The algebra $A$ has norm-controlled inverses, whenever there exists a ...
Tomasz Kania's user avatar
  • 11.3k
3 votes
0 answers
181 views

Completely positive, unital maps acting on unitary operators [solved]

Call a completely positive, irreducible, unital map $E$ on the operators on a finite-dimensional Hilbert-space primitive if there is only a single eigenvalue with modulus 1 (all others have modulus &...
Henrik's user avatar
  • 31
3 votes
0 answers
331 views

About the Moore composition of paths

1) QUESTION (EDIT: 04/28/2020 to remove a possible counterexample) I work with weak Hausdorff $k$-spaces (so all spaces are $T_1$). The internal hom is denoted by $\mathbf{TOP}(-,-)$. Let $\mathcal{G}...
Philippe Gaucher's user avatar
3 votes
0 answers
59 views

Convergence of sesqui-holomorphic kernels on the diagonal

Let $X\subset \mathbb{C}^d$ be a domain. A function (kernel) $K:X\times X\to \mathbb{C}$ is called sesqui-holomorphic if it is holomorphic in the first variable, and anti-holomorphic in the second ...
erz's user avatar
  • 5,529
3 votes
0 answers
146 views

Separating a countable closed set from a point

Let 𝑋 be a separable metrizable space. An intersection of clopen subsets of $X$ is called a C-set. Question. If each singleton of $X$ is a C-set, $A\subseteq X$ is closed and countable, $x\in X\...
D.S. Lipham's user avatar
  • 3,317
3 votes
0 answers
83 views

Reference request for representation theory of TRO

Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading ...
Math Lover's user avatar
  • 1,115
3 votes
0 answers
256 views

How can we solve this kind of saddle point problem?

I'm trying to solve a saddle point problem of the following form: Let $(E,\mathcal E,\lambda)$ be a measure space; $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$ $W$ be ...
0xbadf00d's user avatar
  • 167
3 votes
0 answers
151 views

Completeness of discrete shifts in $\mathbb{R}^n$

Consider the space $L^2(\mathbb{R})$. Let $(x_n)_n \subset \mathbb{R}$ be a sequence and $f \in L^2(\mathbb{R})$ a functions. It is well understood under which assumptions the span of the set $$ S = \{...
Muzi's user avatar
  • 173
3 votes
0 answers
648 views

When the square root of integral operator becomes also integral operator (with continuous kernel)?

Let $X$ be a compact metric space and $\mu$ be strictly positive Borel measure on $X$. Let $T:L^2(X,\mu)\rightarrow L^2(X,\mu)$ be a self-adjoint, compact, and positive operator on the Hilbert space $...
S.Lim's user avatar
  • 469
3 votes
0 answers
133 views

Lower bound on the intersection of $\ell_1$ $n$-balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ in $\ell_1$ norm, with distance $d$ and radius $R$. Is there a lower bound on the volume of the intersection between the two n-balls? (assuming the ...
GWB's user avatar
  • 301

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