All Questions
9,779 questions
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 63.9k
1
vote
0
answers
66
views
The derivative of semigroup in the weak sense imply strong sense
Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
- 775
7
votes
1
answer
184
views
Functional calculus on the Schwartz space instead of $L^2$?
As far as I know, functional calculus is typically carried out on Hilbert spaces with (possibly unbounded) self-adjoint operators.
However, I wonder if there is a way to do it on the space of test ...
- 76
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
-1
votes
2
answers
251
views
$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 12.9k
5
votes
2
answers
432
views
Does closedness of the image of unit sphere imply the closed range of the operator
Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
- 12.6k
1
vote
0
answers
63
views
Is there any other norms besides cut norm defined on graphon?
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
- 359
0
votes
0
answers
66
views
convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
CommunityBot
- 1
2
votes
0
answers
96
views
Isometric Schröder-Bernstein theorem for injective Banach spaces?
It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space.
Let $X$, $Y$ be two injective Banach spaces such that,
...
- 63.9k
7
votes
0
answers
131
views
Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
- 3,425
5
votes
2
answers
517
views
Functions whose product with every $L^1$ function is $L^1$
Let $\mu$ be a probability measure and $f$ a measurable function whose
product with any integrable function is integrable: $$
\int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$
...
- 578
1
vote
1
answer
87
views
Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$
Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to ...
- 3,425
2
votes
0
answers
66
views
interchange of integrals and semigroup without the semigroup being an integral operator
In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears
The formula (1.5.2) is Duhamel formula:
$$u(t) = T(t)u(...
- 677
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 578
5
votes
1
answer
483
views
Can you always extend an isometry of a subset of a Hilbert Space to the whole space?
I remember that I read somewhere that the following theorem is true:
Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
- 63.9k
3
votes
0
answers
104
views
Comparing unitaries which are perturbatively close
Let $\mathcal{H}$ be a Hilbert space and let $H_0$ and $H_1$ be two Hermitian operators on $\mathcal{H}$. Thinking of $H_1$ as a perturbation of $H_0$, the Duhamel formula allows us to write $e^{-...
- 452
3
votes
1
answer
263
views
Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
- 1,190
2
votes
0
answers
137
views
Why a function induced by the infimum of the arclength of curves is Lipschitz?
Recently I have read a paper "Weighted Trudinger-type Inequalities" written by Stephen M. Buckley and Julann O'Shea and published by Indiana University Mathematics Journal in 1999, MR1722194,...
- 6,367
1
vote
0
answers
105
views
Can a uniform weak null set of $c_0$ be uniformly embedded into a Hilbert space?
Remark that a uniform weak null set $A$ of $c_0$ satisfied that for any $\epsilon>0$ there exists a positive number $N(\epsilon)$ such that for every $f$ in $B(\ell_1)$, the unit ball of the ...
- 16.4k
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 4,713
0
votes
0
answers
97
views
Heine-Borel property for (probability) measures on $\mathcal{S}'$?
For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
- 3,477
0
votes
1
answer
150
views
When are infimal convolutions contractions?
Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution
$$
\...
- 1,724
3
votes
2
answers
228
views
Sobolev extension problems of $W^1_\infty(\Omega)$
Recently I have read the paper Whitney's problem on extendability of functions and an intrinsic metric written by Nahum Zobin and published by Advances in Mathematics in 1998. I am confused about one ...
- 3,584
0
votes
0
answers
39
views
Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
3
votes
1
answer
102
views
Literature containing basic knowledge of homogeneous functions
Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
- 63.9k
1
vote
2
answers
622
views
Kähler manifold with Ricci-flat Kähler form
hallo,
I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
- 586
1
vote
1
answer
209
views
Rate of convergence of mollified functions in $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
1
vote
0
answers
63
views
$\operatorname{ker}(q_I \otimes^{\text{min}} q_J) $ is a primal ideal of $\mathcal{A} \otimes^{\text{min}} \mathcal{B}$
In the proof of Theorem $4.1$ of the paper titled continuous bundles of $C^{\ast}$-algebras and tensor products following result is mention with a reference to Proposition $3.3$ of the paper "A. ...
- 1,115
2
votes
1
answer
257
views
Differential equation involving square root
I am absolutely not familiar with differential equations. However, I am facing the following differential equation:
\begin{equation}
a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{y^{2}(x)+d(x)}
\end{equation}
...
- 176
0
votes
1
answer
106
views
Convergence of mollified functions in weighted $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
- 5,005
0
votes
0
answers
43
views
Locally uniformly convexity in kernels (generalized definition of graphon) with cut norm
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
1
vote
0
answers
158
views
Density of Schwartz distributions in the space of distribution
Let $\mathscr S(\mathbf R^3)$ and $\mathscr D(\mathbf R^3 )$ be the space of Schwartz function and test function respectively, $\mathscr S'(\mathbf R^3)$ and $\mathscr D'(\mathbf R^3)$ be their duals....
- 3,065
13
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9)...
- 35.5k
0
votes
0
answers
107
views
Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
- 41
0
votes
0
answers
45
views
Mean value property for fractional laplacian
I just started reading about fractional Laplacian. I am curious on the following questions
Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
- 41
2
votes
2
answers
151
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
- 825
5
votes
1
answer
375
views
Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
- 6,223
4
votes
0
answers
330
views
Book recommendation in functional analysis and probability
I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?
I'm looking for a book that has ...
- 4,713
0
votes
1
answer
151
views
Super-reflexivity is separately determined
I've found this result that states that super-reflexivity is separably determined, i.e., if every separable subspace $Y\subset X$ of a Banach space is superreflexive then $X$ itself is superreflexive. ...
- 2,830
1
vote
1
answer
121
views
Question on complemented subspaces of a product space
Assume that we have closed subspaces $Y_1$ and $Y_2$ of Banach spaces $X_1$ and $X_2$, respectively. If the product $Y_1\times Y_2$ is complemented in $X_1\times X_2$, does it follow that $Y_i$ is ...
- 1,848
0
votes
0
answers
72
views
Domain of a Jacobi operator with unbounded coefficients
Is it possible to describe the domain of a Jacobi operator explicitly?
Let $J$ be the linear operator acting on a real sequence $(u_{n})_{n\in\mathbb{N}}$ by
$$
J(u_{n}) = a_{n+1} u_{n+1} + a_{n} u_{n-...
- 23
2
votes
0
answers
111
views
Everywhere-defined unbounded operators between Banach spaces
In this post, it is said that there are no constructive examples of everywhere-defined unbounded operators between Banach spaces; every example furnished must use the axiom of choice. This seems like ...
- 151
8
votes
1
answer
4k
views
Covering number of Lipschitz functions
What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?
Only 2 results I have found so far are,
That the $\infty$-...
- 1
3
votes
0
answers
318
views
The curse of dimensionality of the Kolmogorov–Arnold neural network
The Kolmogorov–Arnold neural networks (KAN), Ziming Liu et al., KAN: Kolmogorov–Arnold Networks is inspired by the Kolmogorov–Arnold representation theorem (KA theorem). Though it is not proved in the ...
- 2,239
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
3
votes
1
answer
209
views
A few points of clarification on the Martin boundary
Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...
- 2,090
0
votes
0
answers
49
views
Existence of sequence of regular projections
Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
- 869
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 $ℝ^n$, ...
- 11
2
votes
1
answer
148
views
Is projection of a closed subspace Borel?
Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
- 3,584
3
votes
1
answer
228
views
Is compact-open topology stable with respect to injective limits?
Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:...
- 35.5k