All Questions
10,826 questions
5
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2
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258
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Boundary value of Sobolev space
Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
- 3,584
1
vote
0
answers
73
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Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759
5
votes
1
answer
508
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Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)
I've been tackling the following problem for some time,
Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
CommunityBot
- 1
0
votes
0
answers
36
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Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
- 109
3
votes
1
answer
376
views
A more general product rule for weak derivatives?
Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
- 6,367
2
votes
0
answers
57
views
Mappings that preserve local or global minimum
In the most general form, I'm interested in any non-trivial results of the following question.
Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
- 275
1
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0
answers
51
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Compact embeddings RKHSs into Sobolev Spaces
Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
5
votes
1
answer
183
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Question about modular group (Modular theory in operator algebras, section 2.14)
Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20:
I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
- 4,351
2
votes
0
answers
86
views
Besov spaces containing piecewise linear functions
Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
- 63.9k
1
vote
0
answers
72
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Positive eigenfunctions of the discrete Fourier transform
Let $G$ be a finite cyclic group of order $n$ ($n$ need not be prime) and $\mathcal{F}$ the normalized discrete Fourier transform defined on $G$.
Is there a canonical way to construct an eigenfunction ...
- 549
2
votes
1
answer
244
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Characterization of normed spaces based on violation of parallelogram law
For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied.
I would like ...
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6
votes
3
answers
1k
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Additional conditions under which separately continuous implies jointly continuous
Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.
I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):...
- 12.9k
1
vote
0
answers
34
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Discrepancy between probability measures, tested against bounded functions of bounded variance
When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
- 801
3
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0
answers
165
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$S^{n}(V)$ is "approximately" $V$ when $n$ goes to infinity (in the setting of normed space)
Let $B$ be a (separable) Banach space. $(v_{i})_{i}, i\in\mathbb{N}$ being a family of linear independent vectors. $V$ being the span of $v_{i}$. I try to prove that $V$ is dense in $B$. I define a ...
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3
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1
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176
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Question about Lebesgue Bochner spaces
Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.
I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
- 2,670
2
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1
answer
170
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Special function in the Hardy space
Let $H^2(\mathbb{D})$ denote the complex Hardy space, this is: analytic functions defined unit disc $\mathbb{D}$ whose coefficients form a sequence in $\ell^2$. Functions in $H^2(\mathbb{D})$ have a ...
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1
answer
281
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Continuous wavelet transform of a periodic function
I have a question regarding the Continuous Wavelet Transform (CWT) of non finite energy functions, such as $g(t) = a\exp(i\omega_0t)$. We know that the CWT is defined for functions in the Hilbert ...
CommunityBot
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2
votes
0
answers
92
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Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
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1
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1
answer
268
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Best constant for Hölder inequality in Lorentz spaces
It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely
$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...
CommunityBot
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2
votes
1
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117
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Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
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11
votes
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342
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The diagonal operators and unconditionality
The following is well-known:
Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$.
Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with
the ...
- 2,429
2
votes
0
answers
85
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Dirichlet problem for an elliptic operator
consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator.
My question is there is a formula of the ...
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0
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159
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Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 3,065
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2
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792
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How to calculate an integral over the complex unit sphere
We want to calculate the following integral over the complex unit sphere $S^{2n-1}$:
$$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$
where $ z $ is a fixed point in ...
3
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0
answers
58
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Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
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4
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1
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158
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Is the image of a complemented subspace complemented?
This question has been crossposted from mathstackexchange:
Let $X, Y$ be two Banach spaces and $T:X\to Y$ a continuous surjection. Assume $Z$ is a complemented subspace of $X$ and that $T(Z)$ is ...
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59
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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
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3
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1
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177
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Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
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220
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Ultraviolet divergences of entanglement entropy in QFT
I've often read that entanglement entropy in quantum field theory is ill-defined because local algebras are generally of type III, which implies that a trace doesn't exist. For a normal state $\omega_{...
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3
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1
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203
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Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
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4
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1
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309
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On self-duality of non-Archimedean local fields
The question to follow has already been asked by the OP at https://math.stackexchange.com/questions/3454735/on-self-duality-of-non-archimedean-local-fields. Due to a lack of feedback, the OP felt ...
CommunityBot
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3
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0
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167
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Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
2
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1
answer
113
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Showing $\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}\leq \inf _{a \neq 0} \frac{\|\hat{a}\|_{\infty}}{\|a\|}$ in a commutative banach algebra
Suppose $A$ is a commutative Banach algebra, and let $u=\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}$, $v=\inf _{a \neq 0} \frac{r(a)}{\|a\|}$ ($r(a)$ is the spectral radius of $a$). I need to ...
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2
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0
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126
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Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
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225
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Can the trace be computed in any Schauder basis?
I'm cross-posting this question from Math.SE, as it didn't get much attention there.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
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3
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116
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On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
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2
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0
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95
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On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
6
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1
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290
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Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...
- 1,075
4
votes
1
answer
227
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Continuity upgrade for nonlinear maps
Let $E,F,G$ be topological vector spaces such that $F\subset G$ with continuous embedding.
By continuity upgrade I mean the following phenomenon: In some circumstances a continuous linear map $f:E\...
- 1,075
2
votes
0
answers
142
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A $C^*$-algebra with the bidual $B(H).$
Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} \equiv B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$...
3
votes
0
answers
69
views
Perturbation of one-parameter groups of unitary operators
Let $H$ be a Hilbert space and let $h$ be a fixed, densely defined, possibly unbounded, self-adjoint operator on $H$. Letting $B(H)$ denote the space of all bounded operators on $H$, it is well ...
- 2,263
1
vote
2
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156
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Numerical evaluation of monomial divided differences
Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$
I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
- 8,271
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0
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29
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On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
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2
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164
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Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 12.9k
3
votes
1
answer
79
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How to deal with singularities in thin plate splines?
Follow up from this question Thin-Plate-Spline understanding and solution.
In the general case of $\mathbb{R}^N$ the following problem (interpolant which minimizes the Thin Plate Energy, specifically ...
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7
votes
1
answer
253
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Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 2,830
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0
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88
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Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
1
vote
2
answers
209
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Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
- 6,367
1
vote
1
answer
152
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SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
- 24.2k
15
votes
1
answer
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Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
CommunityBot
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