All Questions
3,841 questions with no upvoted or accepted answers
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
3
votes
0
answers
132
views
Takesaki's duality in representation theory of $C^*$-algebras
In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...
- 139
3
votes
0
answers
87
views
Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
3
votes
0
answers
122
views
How are distributions and divergent series summations related?
When we do Fourier analysis, we don't always get convergent series. A classic example comes from considering the Sawtooth function. It has Fourier Coefficients
$$s(x) = \frac{1}{2} + \sum_{n \neq 0} \...
- 1,730
3
votes
0
answers
41
views
Functional of fully nonlinear equations
Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
- 471
3
votes
0
answers
49
views
microlocalisation of a Lagrangian state
Let
$f : \mathbb{R} \to \mathbb{R}$ a smooth function of exponential decay at infinity (I think of something like a Gaussian) ;
$g$ a polynomial of $\mathbb{R}$.
I would like to have a quantitative ...
- 131
3
votes
0
answers
76
views
Absolute continuity of $t \to \lVert u(t) \rVert^2_{H}$ and Gelfand triple : are they equivalent?
Let $V$ be a separable Banach space and $H$ be a separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
and the inclusion maps are continuous with dense images. Here $...
- 3,477
3
votes
0
answers
89
views
Cylindrical Wiener processes or SPDE that can make use of Banach valued rough paths?
Rough paths theory has an often advertised perk that it mostly works for general Banach spaces. I am trying to think of some nice examples that actually use this feature, and am coming up stuck.
The ...
- 193
3
votes
0
answers
83
views
Monotone Characteristic Function
Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$.
What other properties must $X$ have in ...
- 245
3
votes
0
answers
140
views
Trace class operators
There is a notion of trace class operator in a Hilbert space.
Is there a notion of trace class operator in arbitrary Banach space? locally convex space?
A reference will be helpful.
- 21.8k
3
votes
0
answers
240
views
Metrizing pointwise convergence of *sequences* of functionals in a dual space
This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here:
Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...
3
votes
0
answers
111
views
What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
- 2,689
3
votes
0
answers
83
views
Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
3
votes
0
answers
74
views
Locally compact rings with reciprocals
A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
- 974
3
votes
0
answers
84
views
Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$
Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) mapping such that
\begin{equation}
\lVert F(f) \rVert \leq \lVert f \rVert
\end{equation}
for all $f \in L^2(S^1)$. For the space of smooth periodic ...
- 3,477
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
0
answers
94
views
Are U(H) and PU(H) locally uniform topological groups with the norm topology? Towards an instance of infinite-dimensional Hilbert's Fifth Problem
In looking at the work of Enflo generalising Hilbert's Fifth Problem from the Euclidean to the Banach case, there are the following conditions:
the multiplication in the topological group is locally ...
- 35.5k
3
votes
0
answers
126
views
$L^\infty-L^1$ norm of an oscillatory integral operator
Let $\lambda>1$, $x,y\in [0,1]$, and $K_\lambda(x,y)=(1+\lambda|x-y|)^{-2}e^{i\lambda|x-y|}$. Consider the operator
$$T_\lambda f(x)=\int_0^1 K_\lambda(x,y)f(y)dy.$$
We want to precisely estimate ...
- 31
3
votes
0
answers
253
views
Two more topologies on unitary groups
Let $H$ be a separable Hilbert space and let $\operatorname{U}(H)$ be the group of unitary transformations of $H$. It is well known that the weak, strong and compact-open topologies on $\operatorname{...
- 9,853
3
votes
0
answers
131
views
Do all nuclear Frechet spaces have Schauder basis?
When I read an article recently, I see this statement as a conjecture.
So I want to ask, do all nuclear Frechet space have Schauder basis?
Is this still open? Or has been solved? I’ll appreciate any ...
- 61
3
votes
0
answers
98
views
Algebra core for generator of Dirichlet form
This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
- 143
3
votes
0
answers
124
views
Leibniz rule bound for the inverse of the Laplacian?
Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\...
- 3,477
3
votes
0
answers
140
views
Does the Kato-Ponce estimate hold on manifolds?
Recall the Kato-Ponce estimate for fractional powers of the operator $J = (1-\Delta)$,
$$ \| J^s(fg) \|_{L^r} \lesssim \| J^s f \|_{L^{p_1}} \| g \|_{L^{q_1}} + \| J^s g \|_{L^{p_2}} \| f \|_{L^{q_2}},...
- 914
3
votes
0
answers
170
views
Smoothness of height in Manin conjecture
Set up: Let $K$ be a number field. Let $M_K$ be the places of $K$, and define the standard height on $\mathbb{P}^n(K)$ as
$$H([x_0, \cdots, x_n]) = \prod_{v \in M_K} \max\{|x_0|_v, \cdots, |x_n|_v\}$$
...
- 267
3
votes
0
answers
151
views
Reference request: trace norm estimate
In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$
then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 101
3
votes
0
answers
127
views
Number of spatial critical points of a solution to the heat equation in higher dimensions
I would like to know if the number of spatial critical points of a solution to the heat equation can increase. Given $u_0:\mathbb S^n\to\mathbb R$, let $u$ be the solution of the initial value problem:...
- 31
3
votes
0
answers
80
views
Does the Minlos theorem work for real-valued cases as well?
Let $\mathcal{E}(\mathbb{T}^3, \mathbb{R})$ be the Frechet space of real-valued smooth periodic functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
Let us define a real-...
- 3,477
3
votes
0
answers
153
views
Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$
(This question has also been asked on Math StackExchange.)
Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about ...
- 2,830
3
votes
0
answers
86
views
Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$
Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial ,
so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that
$g(|x|^\gamma)$ is positive ...
- 31
3
votes
0
answers
102
views
Can Sobolev space be characterized by spectral decomposition?
Consider a homogeneous Carnot group $\mathbb{G}$ with step $r$. Let $X_1,\cdots,X_m$ be the first layer of its Lie algebra. Denote by $\mathcal{L}=-\sum_{i=1}^m X_i^2$ the sub-Laplacian on $\mathbb{G}$...
- 561
3
votes
0
answers
215
views
Implicit function theorem in Riemannian manifold and Wasserstein space
My question is about to what extent can we extend the implicit function theorem to Riemannian manifolds. In the Euclidean space, consider a bivariate function $F \colon \Theta \times \mathcal{X} \...
- 1,127
3
votes
0
answers
141
views
Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
3
votes
0
answers
60
views
Automatic complete boundedness for bilinear and multilinear maps
$\newcommand{\cb}{\mathrm{cb}}$Let $T : X \rightarrow Y$ be a bounded linear map between Banach spaces. We have the following results concerning automatic complete boundedness:
$\|T : X \rightarrow \...
- 565
3
votes
0
answers
157
views
Growth of the constants from the Stone-Weierstrass Theorem
The Stone Weierstrass theorem for $C([0,1])$ claims that for any continuous function $f:[0,1]\to\mathbb{R}$ and each $n\in\mathbb{N}$, there is a polynomial $p_{n,f}(x)=\sum_ia_{f,n,i}x^i$ such that $\...
- 10.6k
3
votes
0
answers
108
views
Eigenfunction expansion theorem for general manifold for smooth functions
The question starts with the well known facts that: if $f$ is a smooth function on $S^1$, then its Fourier series converges to it in smooth topology.
This must be true in more general setting. I have ...
- 31
3
votes
0
answers
198
views
On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
3
votes
0
answers
111
views
Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...
- 193
3
votes
0
answers
295
views
Dunford-Pettis like properties for Banach spaces of operators
Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...
- 2,605
3
votes
0
answers
115
views
Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
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 $\Omega \times [0,T]$
$$ \partial_{t}u = a_{1}(x,t) \...
- 131
3
votes
0
answers
143
views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
3
votes
0
answers
209
views
Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...
- 71
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
3
votes
0
answers
123
views
Domain of operator which is used in operator monotone function
We are studying the paper
Rupert L. Frank, Leander Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, Journal für die reine und angewandte Mathematik (Crelles ...
- 561
3
votes
0
answers
164
views
Green-Tao's "Polylogarithmic bound for $r_4(N)$"
On P.23 of https://arxiv.org/pdf/1705.01703.pdf, they seemed to suggest that by the non-negativity of $\psi\big(\frac{k}{N}\big)$ for all $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
- 53
3
votes
0
answers
59
views
Sufficient conditions for the weight function to have compact embedding of a weighted Sobolev space
Let $\rho$ be a smooth density function on $\mathbb{R}^N$, that is, $\rho(x)\ge 0$ for all $x\in\mathbb{R}^N$ and $\int_{\mathbb{R}^N}\rho(x) dx=1$. Let $L^p_\rho(\mathbb{R}^N)=\{f: \int_{\mathbb{R}^N}...
- 407
3
votes
0
answers
175
views
Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 204
3
votes
0
answers
390
views
How to prove the polar decomposition of unbounded operators?
Let $ T $ be a closed, densely defined operator on a Hilbert space $ H $. Then there exists a positive self-adjoint operator $ A $, $ D(A)=D(T) $ and a isometric operator $ V:R(A)\to \overline{R(T)} $ ...
- 1,219
3
votes
0
answers
182
views
Rate of uniform approximation by piecewise constant functions
Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...
- 5,405