All Questions
3,859 questions with no upvoted or accepted answers
7
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Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"
For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization
$$
L^p (\mu) = \left\{f : X \to \Bbb{...
- 734
7
votes
0
answers
1k
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Books on von Neumann algebras
I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
- 455
7
votes
0
answers
132
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Different definitions of fractional sobolev spaces
Let $\Omega$ be a bounded and smooth domain in $\mathbb R^d$. For any $s\in (0,1)$ we can define $H_s(\Omega)$ to be the space of functions $u\in L^2(\Omega)$ such that $$(x,y)\mapsto \frac{|u(x)-u(y)|...
- 630
7
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0
answers
3k
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What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
- 1,104
7
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0
answers
187
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distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 167
7
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answers
501
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intuitive connection between The KdV equations and the Virasoro bott group
I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
- 979
7
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0
answers
244
views
Commutation preserving operators
Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...
- 633
7
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340
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Embeddings between weighted Besov spaces
Consider the Besov spaces $B_{p,q}^s(\mathbb{R}^d)$ for parameters $0<p,q\leq \infty$ and $s\in \mathbb{R}$. The weighted Besov space $B_{p,q}^s(\mathbb{R}^d;\mu)$ is defined for $\mu \in \mathbb{R}...
- 2,306
7
votes
0
answers
183
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Is there a quotient of $c_0$ without the approximation property?
The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...
- 71
7
votes
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answers
141
views
Kolmogorov superposition representation of Lipschitz functions
Kolmogorov's superposition theorem states that
if $f:[0,1]^n \to \mathbb{R}$ is an arbitrary continuous function, then it has the representation
\begin{align}
f(x) = \sum_{q=0}^{2n} \Phi_q (\sum_{p=1}...
- 371
7
votes
0
answers
241
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A "slice-map" type problem for symmetric tensors in the square of a nuclear C*-algebra
Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.
Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...
- 25.8k
7
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answers
269
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Approximation in the tensor square of a weakly exact von Neumann algebra
Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
- 25.8k
7
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0
answers
927
views
What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
- 1,051
7
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0
answers
744
views
What function space does holomorphic functional calculus give us?
Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
- 175
7
votes
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answers
559
views
The Banach space of bounded functions with countable support
Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...
- 11.3k
7
votes
0
answers
116
views
Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
- 71
7
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289
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Trace class norms of special integral operators
Let $\mu$ be a finite compactly supported Borel measure on the real line. On the space $L^2(\mu)$ consider the integral operators
$$
(K_a f)(x)=\int k_a(x, y)f(y)d\mu(y)
$$
with
$$
k_a(x, y)=\frac{a\...
- 452
7
votes
0
answers
234
views
Is there a tensor norm that preserves Rosenthal Banach spaces?
By a Rosenthal Banach space I mean one that does not contain an isomorphic copy of $\ell_1$. Is there a tensor norm $\alpha$ such that the Banach tensor product $E\otimes_\alpha F$ is Rosenthal if $E$ ...
7
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0
answers
293
views
Complex interpolation of a Banach space and its antidual when the space has a basis
Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
- 4,461
7
votes
0
answers
2k
views
Prokhorov's theorem for finite signed measures?
Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...
- 339
7
votes
0
answers
237
views
Inferring asymptotic behaviour from the dominant pole of the Laplace transform
Hi,
I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response:
dominant pole in the laplace transform
I have a vector function $X(...
- 349
7
votes
0
answers
199
views
Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
- 1,040
7
votes
0
answers
708
views
Minkowski's Inequality for Integrals in Orlicz spaces
EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
- 882
7
votes
0
answers
299
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
7
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0
answers
355
views
An $L^{\infty}$ version of principal component analysis?
I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
- 193
7
votes
0
answers
266
views
Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
- 879
7
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answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
7
votes
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answers
140
views
integral transforms defined by polygons
Are there any literature on operators of the form
\[ (Tf)(x) = \int K(x,y)f(y), dy\]
where $K(x,y)$ is the characteristic function of a polygon in $\mathbb{R}^2$. I would like to know if the spectrum ...
- 22.8k
7
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0
answers
161
views
Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
7
votes
0
answers
1k
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Reference request: Arzela-Ascoli theorem for smooth Hölder norms
Could anyone suggest a textbook account of the Arzela-Ascoli theorem for $C^{k,\alpha}$ norms?
- 29.1k
6
votes
0
answers
76
views
About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 193
6
votes
0
answers
212
views
Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck
In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
- 61
6
votes
0
answers
162
views
Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
6
votes
0
answers
159
views
Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 721
6
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0
answers
129
views
Are there isospectrally equivalent exotic spheres?
Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
I would be happy ...
6
votes
0
answers
293
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Looking for Mackey's PhD thesis, "The subspaces of the conjugate of an abstract linear space"
I'm looking for a copy of George Mackey's PhD thesis, The subspaces of the conjugate of an abstract linear space (Harvard Univ., 1942), but am currently struggling to find one online, with the only ...
- 11.8k
6
votes
0
answers
147
views
Estimating the Hausdorff dimension of the discontinuity set of a function
Suppose $\sum_{\xi \in \mathbb{Z}^d}{a_{\xi}}e^{i\langle x, \xi\rangle }$ converges spherically pointwise to $0$ for all $x \in \mathbb{T}^d$, i.e. $\lim_{R \to \infty} \sum_{|\xi| < R}{a_{\xi}}e^{...
- 161
6
votes
0
answers
220
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
6
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0
answers
98
views
Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
- 61
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219
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 370
6
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0
answers
145
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A function whose derivatives belong to $BMO(\mathbb{R}^n)$
I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...
- 411
6
votes
1
answer
331
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
- 3,477
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
6
votes
0
answers
253
views
Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
- 25.8k
6
votes
0
answers
187
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
- 61
6
votes
0
answers
217
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Detailed examples of induction on scale
I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight ...
- 265
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
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
6
votes
0
answers
110
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 ...
