All Questions
10,233 questions
2
votes
1
answer
250
views
Norm continuity of the predual of a von Neumann algebra
Let $M$ be a von Neumann algebra and let $(p_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M_{\ast} $, the predual of $M$.
It is well known that
$\| p_i f \|_{M_\ast}\to_{i} 0$
for ...
- 617
3
votes
0
answers
88
views
Using a maximum principle to deduce regularity
Suppose $\Omega \subset \mathbb{R}$ is an bounded domain and that $u \in C(0,T; H^{2}) \cap L^{2}(0,T; H^{3})$ where $T >0$.
Consider the PDE on $\Omega \times [0,T]$
$$ \partial_{t}u = a_{1}(x,t) \...
- 131
2
votes
1
answer
273
views
Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space
Problem Statement
Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
- 53
5
votes
0
answers
265
views
Automorphic Banach spaces
A Banach space $X$ is called automorphic if for every closed subspace $Y\subseteq X$ with $\dim X/Y=\infty$, every automorphism (= linear continuous isomorphism) of $Y$ can be extended to an ...
- 4,535
16
votes
6
answers
2k
views
Finding closed subspaces whose sum isn't closed
Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
- 42.8k
4
votes
2
answers
306
views
Making sense of $1+1$ massless bosonic free field as a "distribution" rather than tempered
The question has been motivated by the fact that the $1+1$ massless bosonic free field suffers the infrared problem as a "tempered distribution".
The reason is essentially that $\int_{\...
- 3,477
2
votes
1
answer
111
views
Lipschitz maps with Hölder inverse preserve the doubling property
Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?
- 5,405
3
votes
0
answers
143
views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
5
votes
0
answers
204
views
Given a totally ordered system of Banach spaces, can we we always change the norms to get isometric embeddings?
Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...
- 609
5
votes
1
answer
353
views
Family of functions with prescribed derivatives
Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
- 4,115
2
votes
1
answer
296
views
An abstract characterisation of weak* topologies
Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if ...
- 153
5
votes
1
answer
169
views
Hadamard factorization of a function in the Fock space
An entire function $F: \mathbb C \to \mathbb C$ belongs to the Fock space $\mathcal F^2$ if
$$
\int_{\mathbb C} |F(z)|^2e^{-|z|^2} \, dA(z) < \infty.
$$
It is well-known that every $F \in \mathcal ...
- 190
6
votes
1
answer
135
views
Small shifts of weakly converging sequences in $L^1$
$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
- 128k
6
votes
0
answers
123
views
Can two eigenfunctions be almost linearly dependent in a region?
Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
- 879
1
vote
0
answers
128
views
Surjectivity of perturbed linear operators
Consider two Banach spaces $X$ and $Y$ and two linear bounded operators $A,B:Y\rightarrow Y$.
Suppose the following:
(1) Y is reflexive (or even uniformly convex);
(2) $X\cap Y$ is dense in $X$ and $Y$...
- 11
4
votes
0
answers
141
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
3
votes
0
answers
209
views
Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...
- 71
4
votes
2
answers
364
views
Equivalence between two Sobolev norms on manifolds
On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.
Use pseudo-differential operators on $M$...
- 71
3
votes
0
answers
82
views
Making a space UMD via interpolation
Recall that a Banach space $B$ has Unconditional Martingale Difference (UMD-$p$) if there is a constant $C_p$ such that for every $B$-valued martingale difference sequences $(d_n)_n$ and choice of $\...
- 408
6
votes
0
answers
188
views
Measurability of eigenvalues-eigenvectors of a positive compact operator
Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.
By the spectral theorem, given $a \in A$, there are scalars $...
- 163
0
votes
0
answers
273
views
Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
1
vote
1
answer
195
views
Concrete example of non-norm-attaining bounded linear operator on disc algebra
A bounded linear operator $T$ from a Banach space $X$ to a Banach space $Y$ is called norm-attaining, if there exists a vector $x\in X$ with $\|x\|=1$ such that
$$\|Tx\|=\|T\|.$$
Let $\mathbb{D}=\{z\...
- 149
5
votes
0
answers
103
views
What conditions guarantee differentiability of higher-order functions?
Sorry if my language is not precise.
Let $f_i:\mathbb{R}^m\to \mathbb{R}^n$ be some differentiable functions in the sense of auto-diff. How can you construct new functions $g(f_1,f_2,f_3,...)$ such ...
1
vote
1
answer
300
views
Isometries of Hilbert space
It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two ...
- 1,361
2
votes
0
answers
119
views
Fréchet and DF spaces
Is there a canonical way to make a DF-space Fréchet while keeping the same vectorial structure? Or the converse? I've been looking in the classical books for locally convex spaces but haven't found ...
- 61
1
vote
0
answers
78
views
Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space
Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...
- 519
0
votes
0
answers
105
views
Is identity map on the space of smooth maps smooth?
I'm curious about the identity map on the space of all smooth maps (between two locally convex topological vector spaces in the sense of convenient calculi) when we equip the space with different ...
- 143
0
votes
1
answer
161
views
Verifying the proof of a bilinear estimate in $L^2$
$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
- 159
4
votes
0
answers
152
views
Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators
I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
- 243
8
votes
2
answers
983
views
What happens if we consider functions of bounded variation that are not in $L^1$?
A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that
$$
\int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx
\leq
C \sup_{ x \in \...
- 5,327
0
votes
0
answers
111
views
Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator
I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...
- 101
1
vote
1
answer
150
views
eigenvalues of matrices (with positive entries)
I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
- 617
0
votes
0
answers
131
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
- 307
1
vote
1
answer
89
views
Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?
Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...
- 825
1
vote
0
answers
82
views
Name for natural norm on functions to non-linear targets
Is there a standard term for the quasi-norm
$$\|f\|_{[k]}=\sum_{i=1}^k(\sup\|f^{(i)}\|)^{1/i}$$
?
It is useful due to the fact that it is reasonably compatible with post-composition by smooth ...
- 18.7k
2
votes
2
answers
755
views
Derivative of the absolute value
Let $f \in W^{1,p}(U)$, then how to prove that $|f| \in W^{1,p}(U)$, where $W$ means the sobolev space over some open subset $U \in \mathbb{R}^n$.
In Lieb's Analysis he prove that Let $f$ be in $W^{1,...
- 23
4
votes
0
answers
132
views
A Lipschitzian's condition for the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
- 291
1
vote
0
answers
52
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
- 287
6
votes
0
answers
111
views
Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
4
votes
2
answers
158
views
A ball with slit at the radius is not $W^{1,1}$-extension domain
Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
- 1,101
4
votes
0
answers
189
views
About the structure of smooth automorphic forms
Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
- 111
5
votes
1
answer
151
views
On existence of a concave function
Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...
- 4,115
2
votes
0
answers
259
views
Research in analysis of PDEs
I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev ...
- 149
1
vote
0
answers
277
views
Fundamental Solution to Biharmonic Equation in 3D
(This is a repost of a question posed in StackExchange that didn't get any replies.)
Is anything known about the fundamental solution to the equation:
$$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$
for ...
- 431
2
votes
1
answer
195
views
Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$
I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
- 229
4
votes
2
answers
199
views
Is the conditional expectation of a Caratheodory function a Caratheodory function?
Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
- 165
2
votes
0
answers
160
views
On Fredholm alternative for Neumann conditions
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases}
-\Delta ...
- 1,350
1
vote
0
answers
71
views
Lipschitz isomorphisms of $C(\omega^\omega+)$
Let $C(\omega^\omega+)$ denote the Banach space of continuous, scalar-valued functions defined on $\omega^\omega+=[0,\omega^\omega]$. Suppose that $X$ is a Banach space and $U:C(\omega^\omega+)\to X$ ...
- 81
2
votes
0
answers
92
views
Linearization stability condition
The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs.
Theorem.
Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
- 319
8
votes
0
answers
135
views
A geometric intuition about convexifiability
I've come up with the following conjecture about convexifiability being determined by "important" sets in Banach spaces. To me, the conjecture looks quite innocuous and intuitive, but I'm ...
- 81