All Questions
3,600 questions with no upvoted or accepted answers
4
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115
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Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$
I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly.
Question 1. In the ...
- 131
4
votes
0
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74
views
Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?
Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
4
votes
0
answers
239
views
The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras
I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in understanding the proof of Proposition $4.5$ from the paper.
Let $A$ ...
- 1,115
4
votes
0
answers
123
views
Extreme points in the space of ucp maps
Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....
- 2,830
4
votes
0
answers
195
views
Bounded cohomology and unitary representations
On page 9 of Nicolas Monod's very nice ICM report "An invitation to bounded cohomology" (https://egg.epfl.ch/~nmonod/articles/icm.pdf), he mentions that bounded cohomology may be related to ...
4
votes
0
answers
495
views
Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175
4
votes
0
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237
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Does SO(n) have Lafforgue's Strong Property (T)?
On page 13 of the monograph of Bekka, de la Harpe, and Valette on Kazhdan's property (T), it is written "for $n \geq 3$, the compact group $\mathrm{SO}(n)$ has the strong property (T)," ...
4
votes
0
answers
334
views
Hodge decomposition on non-compact manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
- 1,171
4
votes
0
answers
152
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Maximally fine topologies on $B(H)$ making the unit ball compact
Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
- 190
4
votes
0
answers
135
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Reverse Sobolev inequality for family of holomorphic functions
Denote by $P_m$ the space of polynomials of degree $m$ in a single complex variable $x$. This is a "Reverse Sobolev inequality":
Theorem. Let $U \subset \mathbb{C}^2$ be an open domain and $...
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4
votes
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140
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Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
4
votes
0
answers
143
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Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators
I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
- 243
4
votes
0
answers
132
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A Lipschitzian's condition for the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
- 291
4
votes
0
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189
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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
4
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0
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310
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Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$
For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...
- 6,853
4
votes
0
answers
100
views
Banach spaces $X$ with $L_2(X)$ isomorphic to $\ell_2(X)$
I've recently come across this interesting thread
Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
I'm interested in the opposite question. Are conditions known that ensure a Banach ...
- 61
4
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60
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Conditions ensuring that the paraproduct remainder is well-defined
In short, my question is: are there conditions that one can impose on two tempered distributions $u$ and $v$ that will guarantee that the paraproduct remainder $R(u,v)$ is well-defined and is "...
- 683
4
votes
0
answers
73
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The embedding of a Banach lattice in an ultrapower
Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...
- 4,461
4
votes
0
answers
134
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Weighted logarithmic Sobolev inequality
$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
- 5,380
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
4
votes
0
answers
103
views
Characteristic of Sobolev space generated by Hörmander vector fields
Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
- 287
4
votes
0
answers
102
views
Geometrical meaning of spaces that possess the weak* uniform Kadec-Klee property
What is the geometric meaning or interpretation of spaces that possess the weak* uniform Kadec-Klee property?
I am writing the last part of my undergraduate thesis and I would like to add a comment in ...
4
votes
0
answers
189
views
Gelfand's transform for noncommutative $C^*$-algebras
Please excuse me if this is well-known, I am not very familiar with the general theory of $C^*$-algebras.
Let $A$ be a unital separable liminal $C^*$-algebra (in the case I am interested in, ...
- 3,446
4
votes
0
answers
132
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Fréchet differentiable non-reflexive Banach spaces
Consider $X$ a Banach space and its continuous dual $X^*$. We know that if the dual norm $\|\cdot\|^*$ of $X^*$ is Fréchet differentiable then $X$ is reflexive (e.g., see Theorem 8.6 in
Fabian, Marián;...
- 105
4
votes
1
answer
311
views
Conormal distributions and the wave front set
Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
4
votes
0
answers
199
views
Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
- 2,818
4
votes
0
answers
109
views
Reflexivity for the compact open topology on a topological $\mathbb{R}$-vector space
On page 20 of these lectures, Peter Scholze and Dustin Clausen show that a broad class of topological vector spaces is reflexive, i.e. $V \cong [[V, \mathbb{R}], \mathbb{R}]$, where we endow these hom-...
user30211
4
votes
0
answers
197
views
Bailey's lemma in number theory
A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by
$$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$
or equivalently
$$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
- 41
4
votes
0
answers
111
views
non centrally free actions of ameanable groups on the hyperfinite III_1 factor
Let $R$ be a hyperfinite $\mathit{III}_1$ factor, and let $Out(R)$ be its set of automorphisms modulo inner automorphisms. There is a canonical and important homomorphism $\phi:\mathbb R\to Z(Out(R))$ ...
- 43.2k
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
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4
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0
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246
views
A question regarding the Hahn-Banach theorem and Banach limits
Set theorists typically prove the existence of Banach limits (EBL) using the Ultrafilter Theorem or, its equivalent, the Boolean Prime Ideal Theorem (BPI). Analysts, on the other hand, typically prove ...
- 6,453
4
votes
0
answers
622
views
Simple constructive proof for the hyperplane separating theorem (HST)?
The Hyperplane Separation Theorem (HST) is usually proved through the existence of a unique minimum-norm vector in a nonempty closed convex set. I think this is an existential proof which applies to ...
- 599
4
votes
0
answers
161
views
Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
- 5,230
4
votes
0
answers
164
views
Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
4
votes
0
answers
81
views
Does this sequence of functions converge in a distributional sense?
Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let
$$\tilde{K}^{\epsilon} := K^{\...
- 469
4
votes
0
answers
120
views
Are fibers in the corona of $H^\infty$ separable?
Let $\mathcal{M}(H^\infty(\mathbb{D}))$ denote the spectrum of the Banach algebra $H^\infty$ and $\mathcal{M}_z(H^\infty(\mathbb{D}))$ the fiber over $z\in \mathbb{D}$, i.e. $\{\varphi\in \mathcal{M}:...
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4
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0
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311
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Estimates of the Frobenius norm of commutator
Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
- 949
4
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0
answers
212
views
"Cyclic vector" of sequence of operators
I recently encountered the following somewhat random-looking problem in my research. At first I thought that should not be too hard, but now, the more I think about it, the more interesting it seems.
...
- 9,853
4
votes
0
answers
116
views
$\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains
I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary,
$$
\left\{
\begin{...
- 213
4
votes
0
answers
160
views
An estimate for the Benjamin-Ono equation from T. Tao's well-posedness paper
In https://arxiv.org/abs/math/0307289 (eq. (8)),
for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$
(where $H$ denotes the Hilbert transform) the following estimate is stated (...
- 839
4
votes
0
answers
132
views
$L_1$-subspace of the predual of a von Neumann algebra
If $M$ is a type $II$ von Neumann algebra, then the predual has a complemented subspace isometric to $L_1(0,1)$. It follows from the existence of expectation. However, I don't know whether such a ...
- 617
4
votes
0
answers
184
views
Weak* HI Banach spaces
The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable. (See the previous posts Decomposable Banach Spaces, Hereditarily ...
- 986
4
votes
0
answers
83
views
Crossed products of A by ℤ: non-stably isomorphic examples
What are some good sources of examples (and/or the simplest example) for:
Pairs of automorphisms $\phi,\psi:A\to A$ over the same base $C^*$-algebra $A$
with non-stably isomorphic crossed products, i....
- 598
4
votes
0
answers
127
views
Can the injective envelope ever be injective for $*$-homomorphisms?
The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
- 3,984
4
votes
0
answers
255
views
Pointwise convergence of kernels of Hilbert-Schmidt operators
Lately I was discussing different types of convergence for Hilbert-Schmidt operators and during that discussion we ended up talking about pointwise convergence of Fourier series. I have already asked ...
- 1,045
4
votes
0
answers
165
views
Tensor product of representations on a compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$.
Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
- 175
4
votes
0
answers
174
views
Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
- 537
4
votes
0
answers
104
views
Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
- 41
4
votes
0
answers
656
views
Eigenvalues of Matérn covariance function
Recall that Matérn covariance function $C_\nu(d)$ is defined as
$$
C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
- 397
4
votes
0
answers
116
views
Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
- 801