All Questions
10,935 questions
2
votes
0
answers
156
views
Fourier multiplier on $L^1$
On the Wikipedia page,
one can read that an iff condition for L1 boundedness of the Fourier multiplier m(D) is that
$$
\hat m\quad\text{ is a Borel measure with finite total mass. }
$$
There is no ...
- 16.2k
63
votes
5
answers
10k
views
Jean Bourgain's relatively lesser known significant contributions
Jean Bourgain passed away on December 22, 2018.
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions,...
3
votes
1
answer
184
views
Question regarding proof of Littlewood-Paley
I posted this question on Math.SE where I unfortunately received no answers even after a bounty. As such, I am putting it here, in hopes to receive a response.
For the proof of Theorem 6.1.6 in ...
- 133
0
votes
0
answers
88
views
Intersection of Sobolev Spaces
Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\...
- 807
5
votes
1
answer
325
views
von Neumann algebra of canonical commutation relations
In quantum mechanics we have position and momentum operators $P$ and $Q$ acting on $L^2(\mathbb{R})$ in the usual way. I'm wondering what the von Neumann algebra generated by the bounded functions of $...
- 439
2
votes
3
answers
457
views
Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
17
votes
1
answer
570
views
Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?
(cross-posted from this math.SE question)
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
- 775
11
votes
2
answers
847
views
Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?
Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality
$$
\...
- 1,141
1
vote
0
answers
210
views
How to show that every Von Neumann algebra is unital?
I was reading the book on operator algebra by Kehe Zhu. The proof of theorem 17.7 (page 107) goes like this :
He first considered the set of all non-empty finite subsets of the set of all projections ...
- 173
0
votes
1
answer
161
views
Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...
- 641
2
votes
1
answer
93
views
Why do distributional isomorphisms preserve joint distribution?
Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}',\mu')$ be probability spaces and
$$f_1,\ldots,f_n:\Omega\to\mathbb R,\; f_1',\cdots, f_n':\Omega'\to\mathbb{R}$$
be integrable random ...
1
vote
1
answer
181
views
Is $\sup_{f\in \mathcal{F}}\left|\int _Xfg \, d\mu\right|<\infty$ true for all $g\in L^\infty _\mathbb{C}(\mu )$?
Suppose that $(X,\mathcal{A},\mu )$ is a finite measure space. Let $\mathcal{F}\subseteq L^1_\mathbb{C}(\mu )$. If $\sup_{f\in \mathcal{F}}\left|\int _Xf\varphi \, d\mu\right|<\infty$ for all ...
- 15
1
vote
2
answers
115
views
Computation of tangent functional
In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows.
If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as
\begin{equation}
\...
- 297
2
votes
0
answers
35
views
Continuity of Kernel Mean Embeddings
Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
- 161
1
vote
1
answer
211
views
Tensor product of faithful normal states is faithful
I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful.
I also ...
- 439
2
votes
0
answers
77
views
Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem
Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
- 809
4
votes
2
answers
378
views
A possible measure-theoretic pathology
Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$
that has the following "shadow property":
For any aligned open square $C = A \times B$ that intersects $S$, ...
- 41
2
votes
0
answers
144
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
- 21
8
votes
2
answers
981
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
7
votes
1
answer
431
views
Reciprocal of a Fourier series (explicit formula for the zero mode?)
Consider a complex Fourier series
$$f(\varphi)=\sum_{m=-\infty}^\infty a_me^{im\varphi}$$
Its reciprocal (where it exists) also admits a Fourier series expansion
$$\frac{1}{f(\varphi)}=\sum_{m=-\infty}...
- 1,035
3
votes
1
answer
167
views
What are the ‘refinements’ in Christ's method of refinements?
I have gathered that Christ's ‘method of refinements’ originated in his paper Convolution, curvature, and combinatorics: a case study to prove the sharp range of $L^p$-improving estimates for the ...
- 679
5
votes
2
answers
708
views
Approximation of Hölder continuous functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
0
votes
1
answer
89
views
$c_0(2^{\kappa})$ does not embed in $\ell_\infty(\kappa)$?
How to prove that $(c_0(2^\kappa),\|\cdot\|_\infty)$ does not embed into $(\ell_\infty(\kappa),\|\cdot\|_\infty)$? Recall that $(c_0(2^\kappa),\|\cdot\|_\infty)$ is the Banach space of all families $(...
0
votes
0
answers
22
views
Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 651
6
votes
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
4
votes
0
answers
260
views
On the predual of the James tree space $\mathit{JT}$
$\newcommand\JT{\mathit{JT}}$The James tree space $\JT$ was the first example of a separable Banach space containing no copies of $\ell_1$ such that its dual space is non-separable. Since $\JT$ admits ...
- 4,461
0
votes
0
answers
55
views
Strong sub-differentiability of an equivalent strictly convex norm
First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
- 85
0
votes
0
answers
94
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
5
votes
1
answer
510
views
A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
1
vote
1
answer
339
views
Analysis of functions over Galois fields
I'm trying to understand how harmonic analysis generalises to functions over finite (Galois) fields.
In particular I'm trying to understand - how to meaningfully map the function to somehow "work ...
- 301
14
votes
2
answers
873
views
Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space?
Background:
It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ ...
- 550
2
votes
1
answer
194
views
Continuity of Moore-Penrose generalized inversion
Any matrix $A\in\mathbb{C}^{m\times n}$ has a unique generalized inverse $A^{\dagger}\in\mathbb{C}^{n\times m}$ with the properties $$AA^{\dagger}A=A,\qquad A^{\dagger}AA^{\dagger}=A^{\dagger},\qquad (...
- 1,093
1
vote
1
answer
180
views
Definition and properties of tangent functional
I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$.
We let $\tau(x, \cdot)$ denote the ...
- 297
1
vote
1
answer
150
views
Is the Boltzmann entropy continuous in the supremum norm?
We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\...
- 825
2
votes
0
answers
232
views
Show that $\mathbb{K}\cong M_{n}(\mathbb{K})$ [closed]
I would like to show the following isomorphy but not sure how to go about this:
$\mathbb{K}\cong M_{n}(\mathbb{K})$
Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism ...
- 119
1
vote
2
answers
181
views
Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$
Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE:
$$\Delta f -\frac{1}{2}h f = 0$$
where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
- 3,535
1
vote
1
answer
118
views
A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 3,477
7
votes
1
answer
331
views
A metric characterization of Hilbert spaces
In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
- 41.9k
2
votes
1
answer
159
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
- 4,153
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 434
2
votes
1
answer
228
views
Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set
$$\tag{1}
\int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
- 809
1
vote
1
answer
117
views
$l^2(L^p)$ Decoupling constant of congruent tubes
Demeter's book Fourier Restriction, Decoupling, and Applications give a principle that one cannot decouple in a direction where the manifold is flat. Which is the below proposition:
Proposition 9.5 ...
- 121
11
votes
1
answer
1k
views
How do people prove $\Gamma$-convergence in more complicated settings?
This is a soft question, I guess. $\Gamma$-convergence is a notion of convergence of functionals so that if $F_n$ $\Gamma$-converges to $F$, then cluster points of $\arg\inf F_n$ are minimizers of $F$....
- 1,904
1
vote
1
answer
95
views
Aggregate function of concave functions
I have two functions $f,g$ form $[0,1]^n$ to $R_{\geq 0 }$ that are concave and monotone.
Given a point $x =(x_1,\ldots,x_n)\in [0,1]^n$, I define $\operatorname{cube}(x) = [0,x_1]\times \ldots \times ...
- 107
0
votes
0
answers
59
views
Series representation of functions
Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let
$$
V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\}
$$
...
- 93
6
votes
4
answers
1k
views
Convex set with no interior contained in hyperplane?
Let $K$ be a convex set in a normed space $X$. Assume that $int(K)=\emptyset$ (norm topology). Must $K$ be contained in some (affine) hyperplane?
It's fairly easy to see that this is true in $ℝ^n$, ...
- 165
3
votes
1
answer
214
views
Convergence of spectrum
Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$.
Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
- 1,211
2
votes
2
answers
135
views
Directed sets of positive elements in noncommutative $\mathrm L^p$ spaces
Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathcal M$.
If $1<p<\infty$ and $E$ is a nonempty subset of $\mathrm L^p(\mathcal M,\tau)_+$ such that
for every $x\...
- 407
1
vote
1
answer
159
views
Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?
Let $X$ be an operator space and $CB(X)$ be the set of all completely bounded linear maps $f: X \to X$. Note that $CB(X)$ becomes a Banach algebra for the composition of operators.
Is the ...
- 175
3
votes
1
answer
339
views
On a Poincaré inequality with weight
Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents.
Is it true that there exists a ...
- 1,263