All Questions
12,935 questions
2
votes
0
answers
75
views
Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
- 969
6
votes
1
answer
285
views
Distinguishing the Besov and Triebel-Lizorkin spaces
Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
- 301
3
votes
2
answers
617
views
A problem about how dominated convergence is used in the analysis of variation
I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
- 809
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
5
votes
2
answers
458
views
Logarithm of a bounded operator
Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that
$
A=\exp L
$...
- 16.2k
2
votes
1
answer
131
views
Gradient flows and particle representations
I was looking into gradient flows and their particle representations, mostly in the context of probability.
A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
- 123
4
votes
1
answer
237
views
Closed-form solution to hyperbolic PDE
Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesgue integrable. Consider the hyperbolic PDE
$$
\begin{cases}
\partial_{x,y}u & = A\...
- 5,405
3
votes
1
answer
79
views
Can a lift satisfy Chen's relation, geometric condition but not be a rough path?
Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties:
\begin{align}
&X_{s,t}=X_{0,t}-X_{0,s}\\
&\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\...
- 1,914
2
votes
1
answer
186
views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
3
votes
1
answer
163
views
Rescaling Fourier coefficients of a continuous function by a bounded sequence
This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
- 404
3
votes
0
answers
103
views
How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
- 1,955
0
votes
0
answers
78
views
Elliptic regularity on manifolds with boundary
If X is a smooth manifold with boundary and of dimension m, and P is an elliptic partial differential operator on X with smooth coefficients, and f is a locally integrable function on X with Pf=0 in ...
- 41
3
votes
0
answers
207
views
Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
- 31
0
votes
1
answer
140
views
Approximating a sequence of tempered distributions "uniformly" by Schwartz functions
This question has been motivated by the post making sense of distributions on the diagonal.
Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f ...
- 3,477
1
vote
0
answers
64
views
embedding spaces of probability measures to function spaces
Let $X, Y$ be Banach spaces. I'm considering a bounded linear functional $g:X\to Y$ and its lift $g_\sharp: \mathcal{P}(X)\to \mathcal{P}(Y)$.
I want to consider the inverse of $g_\sharp$ in some ...
0
votes
0
answers
89
views
Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31
2
votes
2
answers
157
views
"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$
Let $H$ be a separable Hilbert space with the inner product $\langle, \rangle$ and $\{ T_n \}$ be a sequence of unbounded closed linear operators with a common dense domain $D \subset H$ such that $...
- 3,477
0
votes
1
answer
77
views
Decay rate of minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
2
votes
0
answers
132
views
Convergence of integral operators' inverses
Let $K$ be a positive definite integral operator with continuous kernel $K(x,y)$ defined by
$$
Kf(x) = \int_0^1 K(x,y) f(y) \, dy.
$$
Let $K_n$ denote the matrix $K(x_i, x_j)$ with $x_i = i /n$ and ...
- 620
1
vote
1
answer
101
views
Image of a complete topological group under open and surjective map is complete?
A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff.
Question.
Let $G$ be a complete topological group and let $H$ be a topological ...
- 532
0
votes
0
answers
99
views
Dual of closure
Currently I'm studying about abstract interpolation theory for my research. One of the basic ways to construct new interpolation spaces, given an interpolation space $E$ with respect to a compatible ...
6
votes
1
answer
249
views
Syndetic sets and Banach limits: reference request
First of all, let us give a few definitions. Suppose that $A$ is a subset of natural numbers. We say that $A$ is syndetic if there is a constant $M$ such that every set of $M$ consecutive natural ...
- 7,193
5
votes
1
answer
221
views
In what sense does the Laplacian on compact intervals converge to one on all of $\mathbb{R}$?
I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here.
For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ ...
- 3,477
3
votes
0
answers
161
views
Lebesgue measure of the boundary of the positivity set of a function is zero?
Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $C=\{w>0\}$;
$w$ is subharmonic ...
2
votes
1
answer
173
views
Difference in essential spectrum between Schrodinger operators
I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
- 633
0
votes
1
answer
192
views
A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 3,104
0
votes
1
answer
50
views
Norm of a $2$-tuple of operators
Let $E$ be a complex Hilbert space and $K_1,K_2$ are bounded linear operators on $E$.
Let $\omega(K_1)$ and $\omega(K_2)$ be the numerical radius of $K_1$ and $K_2$ respectively. That is
\begin{align*}...
- 1,154
0
votes
1
answer
217
views
About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
- 111
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
2
votes
1
answer
286
views
Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?
Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
0
votes
1
answer
75
views
Derive elliptic maximum principle from weak derivatives
Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that
$$...
- 1,755
3
votes
2
answers
406
views
Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
- 11.8k
2
votes
2
answers
308
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
1
vote
1
answer
171
views
How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?
In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...
- 11
5
votes
1
answer
437
views
Elliptic PDEs in Finance
In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
- 5,405
2
votes
0
answers
56
views
Stability on manifold with boundary
Let $(X,\partial X)$ a smooth Kahler manifold with boundary, i.e. the interior of $X$ is Kahler, Donaldson proved that:
Given a smooth vector bundle $E$ over $X$ such that $E$ is holomorphic over the ...
- 395
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
0
votes
1
answer
53
views
Rate of convergence of the minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
11
votes
1
answer
341
views
Density of linear subspaces in $C(K)$
Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space.
...
- 467
1
vote
1
answer
67
views
Norm of differentiation operator with respect to Gaussian norm
Here is a problem from Luenberger's optimization by vector space methods. I would appreciate steps to proceed.
Let $\mathcal{P}_n\subset\mathbb{R}[x]$ be polynomials of degree at most $n\ge0$. Compute ...
- 125
1
vote
1
answer
125
views
Friedrich's second inequality for functions with zero average
Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
- 31
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 ...
4
votes
1
answer
138
views
Fredholm property of linearization of Floer map
I am reading Audin and Damian's book "Morse theory and Floer homology". In Proposition 8.1.4 which reveals the transversality property of moduli space of solutions of Floer equation, the ...
- 51
1
vote
0
answers
93
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
2
votes
0
answers
70
views
Schauder Frames in nuclear vector spaces
In recent years, the definition of frame has been extended to locally convex topological vector spaces (lcs) (1). In particular, let $X$ be a lcs and $X'$ its dual. A sequence $\big((x_n,y_n)\big)_{n\...
- 655
0
votes
0
answers
106
views
Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
- 809
0
votes
0
answers
97
views
Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
- 39
4
votes
1
answer
288
views
Reference request: Gaussian measures on duals of nuclear spaces
I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...
- 721
7
votes
0
answers
151
views
Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
- 439
1
vote
0
answers
53
views
Reference for Density question
Let $ B $ be a reflexive, separable Banach space and $ p \in (1,\infty)$. Then denote by $L^{p}(B)$ the space of all functions $$ f : \mathbb{R}^{n} \to B $$ with
$$
\int_{\mathbb{R}^{n}} \vert f \...
- 31