All Questions
10,828 questions
7
votes
1
answer
505
views
Is $L_q(X^*)$ complemented in $(L_p(X))^*$?
Let $X$ be a Banach space and let $p\in (1,\infty)$. If $q$ denotes the conjugate exponent to $p$, then $L_q(X^*)$ is easily seen to be isometric to a subspace of $(L_p(X))^*$ via the map $$f\mapsto \...
- 11.3k
7
votes
1
answer
572
views
What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?
Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...
- 1,188
7
votes
1
answer
503
views
Poincaré inequality for curl-integrable functions
Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
- 632
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
- 71
7
votes
1
answer
259
views
Trigonometric polynomials on non-compact and non-abelian groups
I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here.
Hewitt and Ross define trigonometric polynomial on a locally compact ...
- 7,426
7
votes
1
answer
444
views
Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?
Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...
- 4,010
7
votes
1
answer
408
views
Is a C* completion of a nuclear Fréchet algebra a nuclear C* algebra?
I am sure that this is well known in the right places, but: Is the C* completion of a star nuclear Fréchet algebra a nuclear C* algebra? (Suppose that the C* norm is continuous with respect to the ...
- 1,143
7
votes
1
answer
1k
views
Norm concentration of trigonometric polynomials - Uncertainty principle
Hi all,
I am interested in the following question (which is quite similar to one I posed a long while ago): Let $P_{N}(t)=\underset{k=-N}{\overset{N}{\sum}}c_{k}e^{ikt}$ be a unit norm trigonometric ...
- 549
7
votes
1
answer
659
views
Compactness of Sobolev embedding for domains of finite measure
Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero ...
- 4,010
7
votes
2
answers
315
views
Duality between extremal points and extremal maps
Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set
...
- 421
7
votes
2
answers
530
views
The kernel of all invariant means
Let $G$ be a discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g \in ...
- 4,432
7
votes
2
answers
1k
views
Yang Mills gradient/heat flow on 4-torus
The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta ...
- 362
7
votes
1
answer
1k
views
Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
- 73
7
votes
1
answer
538
views
Algebraic topology for nonlinear compact operators
There are analogues of certain basic notions in algebraic topology in the theory of Banach spaces. For example, the Brouwer fixed point theorem generalizes to the Schauder fixed point theorem, and ...
- 6,870
7
votes
2
answers
419
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 217
7
votes
1
answer
329
views
A geometric interpretation of the fractional Fourier transform
I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18):
Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
- 71
7
votes
1
answer
290
views
Square-root lattices: where do they appear?
As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
user491917
7
votes
1
answer
414
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 536
7
votes
1
answer
377
views
Prokhorov theorem on non Polish spaces
It is well known that if $X$ is a Polish space and $\mathcal{F} \subset \mathcal{M}_+(X)$ (the set of finite positive Radon measures on $X$) is uniformly tight and bounded in mass, it is relatively ...
- 401
7
votes
1
answer
389
views
the fractional integration method of the proof of Stein-Tomas theorem?
In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in ...
- 429
7
votes
1
answer
283
views
Kolmogorov superposition on the Hilbert Cube
A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form
$$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...
- 13k
7
votes
1
answer
853
views
Weak*-convergence of signed measures
Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...
- 105
7
votes
2
answers
219
views
Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions
I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...
- 7,789
7
votes
2
answers
997
views
Uniform continuity of heat semigroup
I would like to illustrate my question with an example:
It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup.
It ...
- 536
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
7
votes
1
answer
311
views
Homomorphism to multiplier algebra of groupoid $C^\ast$-algebra
If I have a functor $X\to Y$ between topological groupoids with appropriate Haar measures, such that $X_0 \to Y_0$ is injective and a homeomorphism onto its image, then I should have (or rather, I ...
- 35.5k
7
votes
1
answer
272
views
Simple $C^*$ algebras with invariant subspace property
Edit: According to the valuable comment of Yemon Choi I revise the question by replacing "faithful" with "irreducible".
We say that a $C^*$ algebra $A$ satisfies the invariant subspace ...
- 356
7
votes
1
answer
861
views
Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold
I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
- 159
7
votes
1
answer
624
views
Expectation involving maximum of Gaussian variables
Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
- 1,127
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
7
votes
1
answer
606
views
Weak* continuity of positive parts, again
Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\...
- 42.8k
7
votes
1
answer
747
views
Application of Factorization Theory to Oscillatory Integral Estimates
In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form
$$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^...
- 873
7
votes
1
answer
909
views
Proof of a Fourier pair with Bessel functions?
How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
...
- 71
7
votes
1
answer
665
views
Dominated convergence to characteristic function
Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...
- 81
7
votes
1
answer
306
views
An indicator of a planar subset as an element of a tensor product
Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...
- 452
7
votes
3
answers
713
views
Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?
I'm interested in a question regarding the identification of some duals of quasi-Banach spaces.
However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the ...
user14166
7
votes
2
answers
7k
views
Dual operators between Hilbert spaces: with or without Riesz representation
Let $X$ and $Y$ be Hilbert spaces over the real numbers (so complex conjugation plays no role, and everything will be linear in the strict sense). Let $f : X \rightarrow Y$ be a linear continuous ...
- 5,327
7
votes
1
answer
571
views
Categorical duals in Banach spaces
Near the bottom of the nlab page for Banach space I see "To be described: duals (p+q=pq)".
Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of ...
- 25.2k
7
votes
1
answer
654
views
Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
- 820
7
votes
1
answer
331
views
A metric characterization of Hilbert spaces
In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
- 42k
7
votes
1
answer
361
views
What is the analogue of the Jacobi theta function in the Weyl representation?
It is known (see for example the associated Wikipedia entry) that the Jacobi theta function
$$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$
arises from a certain ...
- 171
7
votes
1
answer
762
views
Feynman-Kac formula for the GFF
The Feynman-Kac formula says that $$ \exp(-t(-\Delta+V(X)))(x,y) = \mathbb{E}_{\gamma(0)=x,\gamma(t)=y}\left[\exp(-\int_0^t V\circ\gamma)\right] $$ where $\Delta$ is the Laplacian on $L^2(\mathbb{R}^n)...
- 396
7
votes
1
answer
403
views
Why are we interested in operators that share a basis of eigenfunctions?
I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I ...
- 185
7
votes
1
answer
355
views
Compactly supported probability measure in high dimensions with fast Fourier decay?
For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...
- 91
7
votes
2
answers
552
views
Cut norm versus $l_1$ norm
Let $K$ be the set of $n\times n$ matrices with zero diagonal entries and such that the sum of all entries is zero.
The cut norm of a $n\times n$ matrix $M$ is:
$$
cut(M) = \sup_{S, T, S\cap T = \...
- 2,772
7
votes
1
answer
1k
views
Products of functions in fractional-order Sobolev spaces
It is well known that $\|fg\|_s \lesssim \|f\|_{s_1} \|g\|_{s_2}$ for functions $f: {\mathbb R}^n \rightarrow {\mathbb R}$ under certain conditions on $s$, $s_1$, $s_2$ (i.e. $s_1$, $s_2 \geq s$ and $...
- 91
7
votes
1
answer
269
views
Derive an orthonormal system by Riesz basis $\{g(\cdot-\lambda_k),\ \lambda_k\in\mathbb R, \ k\in\mathbb Z\}$
Let $\{g(\cdot-k),k\in\mathbb Z\}$ be a Riesz basis, and let $\varphi\in L^2(\mathbb R)$ be a function defined by its Fourier transform
$$\hat{\varphi}(\xi)=\frac{\hat{g}(\xi)}{\Gamma(\xi)},$$
where
$$...
- 297
7
votes
1
answer
1k
views
Chain rule for weakly differentiable functions
Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...
- 459
7
votes
1
answer
1k
views
Inductive/Projective Limits of Topological Algebras
It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...
- 71
7
votes
1
answer
537
views
Multivariate Maximal Hilbert Transform
One way to define the maximal Hilbert transform of a function, $f$, is by
$$\mathcal{H}[f](x):=\sup_{\varepsilon>0} \left| \int_{|x-t|\geq\varepsilon} \frac{f(t)}{x-t} \, dt\right|, \quad x\in\...
- 228