All Questions
3,629 questions with no upvoted or accepted answers
1
vote
0
answers
142
views
Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces
I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow.
LANGUAGE
TVS = topological vector space. Any subspace of a ...
- 603
1
vote
0
answers
149
views
BMO estimates of singular integral operators on torus
I have the following elliptic problem:
$$ \Delta u = \operatorname{div}\operatorname{div}S, $$
where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$...
- 123
1
vote
0
answers
21
views
Regularity of solutions of a 2nd order singular integro-differential operator
I have trouble finding the regularity of the solutions to a particular equation. I define
$$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
- 41
1
vote
0
answers
87
views
Continuous choice of null directions for a family of bilinear forms
Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
- 2,095
1
vote
0
answers
54
views
Closed linear span of compact open subsets of a spectral space
Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
...
- 2,605
1
vote
0
answers
48
views
Notation for dominating (or uniformly bounded) function
While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function.
A situation like this. For some true function $f:\mathbb{R} \to \...
- 284
1
vote
0
answers
150
views
Analyticity of solutions to Schrödinger's equation
Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
- 439
1
vote
0
answers
61
views
When is the metric on a Fréchet space homogeneous
Let $(F,d)$ be a Fréchet space over $\mathbb{F}\in \{\mathbb{C},\mathbb{R}\}$. Are there conditions under which, there exists some $C,d>0$ such that: for every $f\in F$ and every $k\in \mathbb{F}$ ...
- 5,405
1
vote
0
answers
33
views
Limiting absorption principle for higher powers of resolvents
Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator ...
- 141
1
vote
0
answers
64
views
Characterization of elements of Hardy Space
Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that
$$
\forall f\in H^2(\partial\...
- 63
1
vote
0
answers
353
views
$H^{1/2}(0,1)$ and continuous functions
It is known that $H^s(0,1)$ embeds into Hoelder continuous functions for $s>1/2$. I am not interested in Hoelder continuity, but merely in continuity: do I get the continuous embedding $H^{1/2}(0,1)...
- 235
1
vote
0
answers
361
views
Operators for norm for some classes of integral operators
Let $T:L^2(\mathbb{R}^n,\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)$ be a linear operator of the form: for all $x\in \mathbb{R}^n$
$$
T_{\kappa}(f)(x):=\int_{u\in\mathbb{R}^n} \, f(u)\...
- 5,405
1
vote
0
answers
106
views
A locally convex $C^*$ algebraic structure on the disk algebra
A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra ...
- 356
1
vote
0
answers
48
views
Question about higher order mean field equation $\left(-\Delta_{g}\right)^{m} u+\lambda=\lambda \frac{e^{2 m u}}{\int_{M} e^{2 m u} d \mu_{g}}$
I'm reading Dr.Luca Martinazzi's paper
Existence of solutions to a higher dimensional
mean-field equation on manifolds
which proves that for $m \geq 1$, there is an existence result for the equation
$$...
- 809
1
vote
0
answers
99
views
Module homomorphisms modulo compact operators
Let $A$ be a Banach algebra. Let $L_a,R_a:A\to A$ denote the left/right multiplication operators $$L_ax = ax, \hspace{5mm} R_ax = xa$$ for all $a,x\in A$. Assume that no nonzero $L_a$ and $R_a$ is a ...
- 2,605
1
vote
0
answers
418
views
Conditions for equivalence of RKHS norm and $L^2(P)$ norm
Let $K$ be a psd kernel on an abstract space $X$ and let $H_K$ be the induced Reproducing Kernel Hilbert Space (RKHS). Let $P$ be a probability measure on $X$ such that $H_K \subseteq L^2(P_X)$ and ...
- 6,853
1
vote
0
answers
73
views
Straightening a function supported on a strip
Given a positive smooth function $b:\mathbb{R}^{N+M}\to \mathbb{R}$ which vanishes exactly outisde of some $U\times\mathbb{R}^M$ ($U\subset \mathbb{R}^N$ open), is there another positive smooth ...
- 151
1
vote
0
answers
97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
1
vote
0
answers
61
views
Primes as the extrema of a functional
I'd like to write down a functional on sequences for which the prime numbers are an extrema.
One generally thinks of the natural numbers first as an ordered set, and then you discover unique ...
- 571
1
vote
0
answers
449
views
Bound on covering number of Lipschitz functions – missing part in proofs of Kolmogorov et al
Given a metric space $(\mathcal{X},\rho)$ and $\mathcal{A}\subset\mathcal{X}$ totally bounded, i.e. $\mathcal{A}$ has a finite $\varepsilon$-covering for any $\varepsilon>0$. Consider $\...
- 11
1
vote
0
answers
98
views
Weak convergence of measures and compact sets
Let H be a separable Hilbert space. Let L be a Hilbert-Schmidt operator from H to H, such that its image is dense in H. This allow one to take the basis of eigenfunctions of $L$: $\{\phi_i\}_{i=1}^\...
- 63
1
vote
0
answers
45
views
Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
- 5,405
1
vote
0
answers
135
views
Description of state space of $C(K,M_n)$?
Edit: closed convex hull added.
I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space.
My guess would be that these are the closed convex hull of states on $C(...
- 598
1
vote
0
answers
85
views
Interpolation between projective and injective spaces
Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex ...
- 1,879
1
vote
0
answers
178
views
A locally convex $C^*$ algebra without zero divisor
Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. Moreover the topology on $A$ ...
- 356
1
vote
0
answers
88
views
2-positivity to 3-positivity
Let $B\in M_3(\mathbb{C})$ and $S_3=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{pmatrix}
$. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$...
- 231
1
vote
0
answers
83
views
What is lost after RKHS embedding of the L1 space?
We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
- 43
1
vote
0
answers
128
views
Self-ajointness of the Laplacian over a Riemannian manifold with boundary
I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).
Let
$(M,g)$ be a Riemannian manifold with boundary;
$E\to M$ be an hermitian fiber bundle;
$\Delta$ ...
- 141
1
vote
1
answer
329
views
Hölder continuity of Radon transform of smooth function
Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
- 6,853
1
vote
0
answers
254
views
Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 5,405
1
vote
0
answers
188
views
$C^0$ norm is bounded by $L^{14}$ norm
Let $M$ be a closed manifold of dimension $6$, and we look at the collection of smooth functions on $M$ which satisfy:
$$
\|f\|_{C^0}\leq C\big(\|f\|_{L^{14}}^2+1\big)
$$
for some fixed $C>0$. Can ...
- 954
1
vote
0
answers
111
views
Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
1
vote
0
answers
63
views
On the strong convergence of generators and the corresponding semigroups
Let $(E,\|\cdot\|)$ be a separable Hilbert space on $\mathbb{R}$. We consider a sequence of strongly continuous semigroup $\{T^{(n)}\}_{n=1}^\infty$ on $E$ [of course, for $n \in \mathbb{N}$ and $t>...
- 721
1
vote
0
answers
748
views
Notation for the space of eventually-zero sequences
An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
- 153
1
vote
0
answers
152
views
Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
- 839
1
vote
0
answers
292
views
Developing measure theory through $\delta$-rings
I am scratching my head trying to understand the notion of measurability in Fell's and Doran's book: Representations of (star)-algebras, locally compact groups, and Banach (star)-algebraic bundles.
...
- 155
1
vote
0
answers
112
views
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
- 1,101
1
vote
0
answers
97
views
Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$
I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
- 452
1
vote
0
answers
100
views
Question about Dirac operator
Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that
$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$
for $\...
- 563
1
vote
0
answers
34
views
Minimum eigenvalue of normal matrix with polynomial basis
For each $n\in \mathbb{N}\cup\{0\}$, let $x_n(t)=\frac{t^n}{n!}$ for all $t\in [0,1]$. As the functions $X_N=(x_0 ,\ldots, x_N)$ are linearly independent, the matrix
$
\int_0^1 X_N(t)^\top X_N(t)\,\...
- 503
1
vote
0
answers
597
views
What is $T T^*$ argument?
During my studying of many papers, some authors used what so-called $T T^*$ argument. I have no clue about this concept (or mathematical tool). Could you please enlighten me with some explanations or/...
- 159
1
vote
0
answers
100
views
$L_1$ convergence rates for multivariate kernel density estimation
Let $X$ be a random variable on $\mathbb R^d$ with probability density function $f$, and let $X_1,\ldots,X_n$ of $X$ be $n$ iid copies of $X$. Given a bandwidth parameter $h=h_n > 0$ and a kernel $...
- 6,853
1
vote
0
answers
142
views
Uniformly continuous semigroups are analytic
Reposting from stackexchange.
I know that every analytic $C_0$-semigroup is differentiable and then every differentiable semigroup is norm continuous.
I want to know where uniform continuity fits in ...
- 131
1
vote
0
answers
94
views
Less strict holomorphy
Using the Grünwald-Letnikov definition of fractional derivative for complex-valued functions, can there be complex function $f(z)$ over all plane, that is not holomorphic but there is $r \in (0,1)$ s....
- 117
1
vote
0
answers
130
views
The functional calculus on continuous functions
Let us consider $C[0,1]$ the space of continuous functions on the closed unit interval. For a given $x$ in $C[0,1]$, let us consider $A(sp(x))$, all analytic functions on a neighborhood of $sp(x)$.
...
- 4,058
1
vote
0
answers
153
views
Extreme points of the unit ball of bounded operators on $L^p(\mathbb{R}_+)$
Is it known what the set of all extreme points of $B(L^p(\mathbb{R}_+))$ (bounded operators on $L^p(\mathbb{R}_+)$) is?
- 11
1
vote
0
answers
82
views
Extreme case of K-interpolation
Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space
$X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
- 319
1
vote
0
answers
57
views
How to show the solution map of NLS is not smooth?
Let $u(\delta, t)$ satisfy
$$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$
Note that the mapping:
$$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$
...
- 325
1
vote
0
answers
97
views
Reference for unique classical solution to quasilinear uniformly parabolic PDEs
In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...
- 11
1
vote
0
answers
92
views
Target space of Green's operator on $L^p$-differential forms on closed manifolds
Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...
- 213