All Questions
3,841 questions with no upvoted or accepted answers
3
votes
0
answers
84
views
"Weakly" nuclear operators (terminology)
Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name.
Let's say that a linear map $T:V\to W$ between locally convex topological ...
- 81
3
votes
0
answers
125
views
Green operator of elliptic differential operator and radius of convergence
Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
- 1,383
3
votes
0
answers
67
views
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\...
- 31
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
$$
...
- 455
3
votes
0
answers
102
views
Determining what happens to the spectrum of Schrödinger operator as boundary condition changes
I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...
- 251
3
votes
0
answers
87
views
Doubt when calculating the S-transform of Hida differential operator
Assume we have a Hida test function $\varphi\in (\mathcal S)$, and $y\in \mathcal S'(\mathbb R)$. Define the Gateaux directional derivative of $\varphi$ (in the direction of $y$) by:
$$D_y\varphi(x):=\...
- 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 ...
- 1,263
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 ...
- 2,281
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....
- 20.2k
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 $\varphi_n:\overline{X}\to X$...
- 5,529
3
votes
0
answers
309
views
The $2\pi$ factor in the Fourier transform and dimensional analysis
I have been thinking about the $2\pi$ factor in the various conventions of the Fourier transform. For example, I was looking for a way to justify the following:
$(*)$ If we define $\hat f(\xi) = \int ...
- 613
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,...
- 1,069
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 (...
- 778
3
votes
0
answers
156
views
The proof of Tomas's restriction theorem $\|\hat{f}(\theta)\|_2\lesssim \|f\|_p$
I am trying to understand the proof of Tomas's theorem:
The proof reads
My question:
How do we get the estimates
$$\|T_k\ast f \|_{\infty}\lesssim 2^{-(n-1)k/2}\|f\|_{1},\qquad\qquad (1)$$
$$\|T_k\...
- 143
3
votes
0
answers
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 (...
- 429
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, ...
- 1,175
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$ ...
- 5,407
3
votes
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 ...
- 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 ...
- 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:
...
- 167
3
votes
0
answers
79
views
Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$
It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...
- 489
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 \...
- 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 ...
- 943
3
votes
1
answer
489
views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
- 751
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 ...
- 1,879
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\...
- 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.
...
- 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 ...
- 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 ...
- 53
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*}
...
- 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(...
- 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 $\Omega \subset \mathbb{R}^n$ ...
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 $T^{*}X$ ...
- 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 $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|...
- 355
3
votes
0
answers
109
views
Integral kernel of composition of linear operators acting in $L^2(\mathbb{R})$
Let $A$ and $B$ be linear operators acting on $L^2(\mathbb{R})$ or some dense subset of that space.
We assume that they are integral operators (possibly with distributional kernel)
$Af(x)=\int_{\...
- 31
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$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 ...
- 375
3
votes
0
answers
237
views
Reconstructing a sine wave using square waves and Möbius inversion: L² convergence?
Let $s$ be the (“square wave”) $1$-periodic real function such that $s(x) = 1$ if $0<x<\frac{1}{2}$ and $s(x) = -1$ if $\frac{1}{2}<x<1$ (and maybe $s(0)=s(\frac{1}{2})=0$ for the sake of ...
- 32.5k
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 ...
- 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 ...
- 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 &...
- 31
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 ...
- 5,529
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 ...
- 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 ...
- 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 = \{...
- 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 $...
- 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 ...
- 301
3
votes
0
answers
70
views
Bounding the norm of Sobolev extension operator
If $\Omega$ is a sufficiently nice bounded open set in $\mathbb{R}^d$, it's known that there exists a continuous linear operator $$\mathcal{E}:W^{1,p}(\Omega)\rightarrow W^{1,p}(\mathbb{R}^d)$$ such ...
- 660
3
votes
0
answers
161
views
Fantappie transform(ation)s in Gelfand et al. "Generalized functions"
In the 6-volume "Generalized functions" a treatment of Fantappie transformations is promised in Vol. 1 (bottom of p.461 of the Russian edition) to come in Vol. 5. However, there is no Fantappie
...
- 14k
3
votes
0
answers
95
views
Sparse perturbation
Let $x, x_0\in\mathbb{R}^n$ be two vectors satisfying $$\frac{\|x\|_1}{\|x\|_2}\leq\frac{\|x_0\|_1}{\|x_0\|_2}.$$
$\| \cdot\|_1$ and $\| \cdot\|_2$ are the $\ell_1$ and $\ell_2$ norm in $\mathbb{R}^n$,...
- 31
3
votes
0
answers
150
views
What is a $C^\infty$ diffeomorphism from $\ell_2\setminus\{0\}$ to $\ell_2$ which is the identity outside a ball?
Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is ...
- 2,041