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Uniform bound on $\lVert \chi_{\{u_n=0\}}\rVert_{W^{s,p}(\Omega)}$ for a bounded sequence $u_n$ in $H^1_0(\Omega)$?

Suppose I have a sequence $u_n \to u$ in $H^1_0(\Omega)$ on a smooth and bounded domain. For some $p>1$ and $s \in (0,\frac 12)$, is it possible to estimate the norm of the characteristic function ...
StopUsingFacebook's user avatar
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Coincidence Topologies for $L^p$ spaces

If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...
ABIM's user avatar
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142 views

Mackey topology

Recall that for a Hausdorff locally convex space $X$ the Mackey topology $\tau (X^*,X)$ is the topology in its topological dual $X^*$ of uniform convergence on all weakly compact absolutely convex ...
Wer Wer's user avatar
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70 views

Limit of spectral projection of increasing sequence of positive operators

Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...
A beginner mathmatician's user avatar
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Link between eigenvalues of a symmetric matrix and a functional space

Let $f_1,\dots,f_n \in L^2(\mathbb{R},\mathbb{R})$ be $n$ mutually orthogonal functions with $\int f^2_i =1$ such that $|\{x \in \mathbb{R} | f_i(x) = 0\}| = 0$ for any $i \in \{1, \dots,n\}$. Does ...
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Sine-Gordon transformation and functional integrals

In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, ...
JustWannaKnow's user avatar
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145 views

“Chapman-Kolmogorov”-convolution vs. smoothness

Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
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135 views

Reference for discrete Laplacian on $\mathbb{Z}$

For $x\in \mathbb{R}^\mathbb{Z}$, let the discrete Laplacian be defined as \begin{align*} (\Delta x)_k = 2x_k-x_{k+1}-x_{k-1}. \end{align*} I am looking for good references about its spectrum (or ...
I love pineapple coffee's user avatar
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546 views

Orlicz–Sobolev spaces

Let $A$ be an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty. $$ We denote by $\widehat{A}$ an N-function equal to $A$ near infinity and $\widehat{...
deval si-mohamed's user avatar
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Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$

The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas. Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
Sanae Kochiya's user avatar
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255 views

Span of a nonlinear function

Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...
ecstasyofgold's user avatar
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191 views

Canonical embedding of Hilbert space in random $L^2$

This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...
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Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$

Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that $$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...
Trần Quang Minh's user avatar
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Smooth sections of finite dimensional bundle and covering space

Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
Aleksandr Alekseev's user avatar
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Limit of balls in $L^p$

Setup: Let $\mu$ be a measure on a measurable space $(X,\Sigma)$, such that for every $p ,q\in [1,\infty)$, $L^p_{\mu}(\Sigma)\subseteq L^q_{\mu}(\Sigma)$ if $p\geq q$. Furthermore, the inclusions ...
ABIM's user avatar
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On mixing and weak mixing subalgebras of finite von Neumann algebras

Let $M$ be a full $\mathrm{II}_1$ factor. Consider mixing and weak mixing subfactors $B$ and $C$ of $M$. Are $B$ and $C$ full?
sibani's user avatar
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Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$

Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$,...
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Special kind of translation and rotational invariance of the numerical range

Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true? If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then $W(B\otimes (X+I_n))\subseteq W(B\...
Piku's user avatar
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On prime factors

Let $M$ be a prime $\mathrm{II}_1$ factor. Let $N$ be a non hyperfinite finite index subfactor $N$, is $N$ prime factor?
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Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?

$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that. For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
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91 views

Does $L^1$ convergence preserve the regularity of this sequence of functions?

Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges $...
W. Volante's user avatar
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113 views

References for a proof or interpretation of deficiency indices theorem (von Neumann)

I am looking for a proof or some interpretation around why the domain of the new extension $D(A_U)$ in the Theorem below is given by its specific formula. I have already searched in papers and here ...
curiosity96's user avatar
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110 views

On an application dominated convergence theorem in vN algebras

$M$ be a $\mathrm{II}_{1}$ factor equipped with the faithful normal trace $\tau$ in the standard form. Let $\tau(Jx'J\eta)=0,\forall x' \in M'$ and fixed $\eta$ in $L^{1}(M,\tau)$. Is it true $\eta=0$?...
user136400's user avatar
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Help showing F is weakly lower semicontinuous

Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...
R. N. Marley's user avatar
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homomorphisms into tensor product algebras

Given a decomposition $H=H_1\otimes H_2$ of a Hilbert space $H$ into the tensor product of the Hilbert spaces $H_1$ and $H_2$ and a *-isomorphism $U: B(H_0)\to B(H)$, where $H_0$ is another Hilbert ...
Arnold Neumaier's user avatar
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32 views

Spectral measures of a family of parameter-dependent self-adjoint contractions on an $L^2$-space

I have a self-adjoint linear contraction $A_g$ on an $L^2$-space of the form $$A_gf=\int\gamma(f,g),$$ where $\gamma$ is Lipschitz continuous and $g$ is an a priori fixed function. Assuming $1-A_g$ is ...
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Is it possible that the dimension of the intersection of a nested sequence of Hilbert space is 1?

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$ Let $\{h_n\}_{n \in \mathbb{N}} \in H$ be a sequence of linearly independent vectors in $H$ Let $$ V= \bigcap_{n=1}^\...
Matey Math's user avatar
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Partial well-posedness results on Schrödinger operators?

Set $ A_i:= -\Delta + V_i :H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3), \ i =1,2 $, where \begin{equation*} V_1 = 0, \ \ (\textrm{No interaction}) \\ V_2 = - \frac{\gamma}...
Yidong Luo's user avatar
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88 views

On invertibility of ergodic averages

Let $x$ be invertible unbounded operator affiliated operator to the $\mathrm{II_{1}}$ factor $(M,\tau)$. Under which condition on $x$, the iterates also $1+\sigma(x)+\cdots+\sigma^{n}(x)$ are ...
sibani's user avatar
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supermodular independent product

For which choices of two supermodular capacities, the independent product of them (given by the formula $w(A\times B)=\mu(A)\cdot \nu(B)$), also is supermodular ? Even particular/concrete cases of ...
George's user avatar
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86 views

Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?

We know that when $I$ is a bounded interval and $1<p\leq \infty$ that the injection $W^{1,p}\subset C(\overline{I})$ is compact. The proof of this fact uses the Arzela-Ascoli theorem on the unit ...
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$ 0 $ locates in the continuous spectrum of Schrodinger operators?

This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range. For ...
Yidong Luo's user avatar
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127 views

On examples of action of C-star simple group on von Neumann algebra

Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
user136400's user avatar
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0 answers
41 views

reference request: non-negative quadratic function on a subspace, extended by infinity outside

I am looking for a reference for some objects that naturally appeared in my research. Suppose $W$ is a subspace of a real vector space $V$, and $f:W \to \mathbb R$ is a non-negative quadratic function ...
user105346's user avatar
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0 answers
113 views

Extrapolate an Interpolation scale

Suppose $X$ and $Y$ are real Banach spaces with a continuous embedding $X\subset Y$. For given $0<\theta<1$ I am interested in constructing using the norms of $X$ and $Y$ a (Quasi-) Banach ...
Philip's user avatar
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0 answers
45 views

Skorohod Space with $J_1$ topology homeomorphic to Frechet Space

Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
ABIM's user avatar
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Existence of the eigenvalue of the dual operator of the transfer operator

In the passage that I marked in green apparently the author uses a relationship between fixed point and eigenvalues. The result that I know of to ensure the existence of this eigenvalue requires that ...
Ilovemath's user avatar
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Integral of kernel in RKHS with respect to probability measure

I'm trying to understand how to work with the expression $$ \int u (d\mu-d\pi)$$ where $u\in \mathcal{H}$, an RKHS. $\mu$ and $\pi$ here are two different probability measures. Since $\mathcal{H}$ is ...
Kashif's user avatar
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0 answers
79 views

Hausdorff distance restricted to linear subspaces

Let $V$ be a Hilbert space, $Q \subset V$ be convex and compact and $Q_n \subset V$ be convex and compact for $n\in \mathbb{N}$ such that $Q_n \rightarrow Q$ for $n\rightarrow \infty$ in Hausdorff ...
Steve's user avatar
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0 answers
149 views

Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold

Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
archer's user avatar
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55 views

Smooth compactly supported function with good scaling with respect to the fractional Laplacian

Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user avatar
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74 views

Parseval type lower bound on sum of squares of function projections

This is a followup to this earlier question Let $f:\mathbb{Z}\rightarrow \{\pm 1\}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it ...
kodlu's user avatar
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479 views

What are the sets on which norm-closedness implies weakly closedness?

Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
Red shoes's user avatar
  • 369
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44 views

S-familiy induced by an operator induces a Schwartz function

Let $T:S(\mathbb{R}^d)\to S(\mathbb{R}^d)$, a continuous linear operator, where $S(\mathbb{R}^d)$ is the Schwartz space. There is a result that guarantees that the family $F=\{\delta_s\circ T\}_{s\in\...
ksoriano's user avatar
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0 answers
224 views

Show convergence of a sequence of resolvent operators

Let $E$ be a locally compact separable metric space $(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$ $E_n$ be a metric space for $n\in\mathbb N$ $(\...
0xbadf00d's user avatar
  • 167
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0 answers
86 views

Characterzing compact actions on von Neumann algebra

Suppose $G$ is a countable discrete group acting on vN algebra $M$, the action is compact. Can we have a topology on Aut$(M)$ such that $\{\sigma_{g}\in \text{Aut}(M):g \in G\}$ form a compact subset ...
user136400's user avatar
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97 views

Does $L^p$ contractivity imply $L^p$ dissipativity?

Does $L^p$ contractivity of an operator semigroup imply the $L^p$ dissipativity of the operator ? Thank you in advance !
siki's user avatar
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54 views

On cyclicity of a module

Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
user136400's user avatar
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68 views

On existence of sequence of unitaries in $II_{1}$ factor $M$

Let $M$ be a $\mathrm{II}_{1}$ factor acting on $L^{2}(M, \tau)$ in standard form, let $\{e_{n}:n \in \mathbb{N}\}$ be fixed orthonormal basis of $L^{2}(M, \tau)$, does there exist sequence of ...
user136400's user avatar
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78 views

$p$-summing operators space is a Banach space

Let $X,Y$ be Banach spaces and $p\geq 1$. A bounded linear operator $T$ is called $p$-absolutely summing, if there is exist $K>0$, such that for all $n\in N$ and $x_1,\dots, x_n\in X$: $$ \left(\...
Kostas's user avatar
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