All Questions
3,573 questions with no upvoted or accepted answers
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123
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On the operators from $l_{p}$ into Tsirelson's space $T$
Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
- 3,343
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54
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Left introversion operators associated to function spaces on semigroups
I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
- 1
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0
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75
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On characterizations of $p$-integral operators
In most of papers and classical text books, $p$-integral operators are defined and characterized by operators and commutative diagrams. This is quite different from $p$-summing operators and $p$-...
- 3,343
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371
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Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
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510
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Composition of upper semi-continuous real valued function with upper semi-continuous matrix valued function
Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if
$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$
where $\preceq$ ...
- 1,590
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115
views
When do block sequences yield disjoint subspaces?
Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to $(...
- 3,115
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63
views
The union of weighted compact supported continuous function
Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
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107
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Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?
Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...
- 123
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188
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One parameter family differentiable dependence for linear parabolic pde's
Consider for example, the Black Schole's equation
$$
\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0
$$
on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The ...
- 383
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173
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Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
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54
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Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$
Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...
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144
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Unitarizability of group representations
Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...
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407
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What does the Plancherel theorem say about positive-definite distributions?
I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.
The ...
- 3,799
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109
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solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,385
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266
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Maximizing Expected Utility
I am currently trying to solve a maximization problem given by
$\max_{f(x)} \int_0^1 \int_\mathbb{R} (c-y\cdot f(x)-d\cdot (x+f(x)-b)^2) \ h(x) \ dx \ dy$.
Or in other words, I have a utility ...
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137
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Heat asymptotics
Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
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358
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Boundedness of heat semigroup on $L^1(\Omega)$
On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...
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187
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About equivalence of two fractional Sobolev/Hilbert spaces
Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space
$$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac 12}|(u,\varphi_k)_{L^...
- 251
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377
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Densely-defined operator with closed range: conditions for operator closed
Suppose we have Banach spaces (or Hilbert spaces) $X$ and $Y$,
and a densely-defined linear operator $A : \operatorname{dom}(A) \subset X \rightarrow Y$ that is densely-defined and with closed range. ...
- 5,327
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150
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smoothness of green's function in wave equation
I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...
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81
views
Approximation property of Fréchet if range is restricted to an embedded Hilbert space
Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...
- 320
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201
views
Range of a trace preserving completely positive projection
I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...
- 515
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256
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Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325
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114
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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104
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Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?
The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...
- 1,771
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127
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Approximation Property: Characterization
Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...
- 598
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156
views
Compact Approximation
This thread originated from MSE: Compact Approximation
This is meant as lemma for: Approximation Property
Given a Banach space $E$.
Denote compact domains by $\mathcal{C}$.
Denote compact ...
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351
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Existence of a complementary closed subspace extending a given subspace
Let $H$ be an infinite dimensional (separable) Hilbert space (or any infinite dimensional Banach space in which every closed subspace has a complementary subspace). Suppose that $X$ and $Y$ are closed ...
- 3,115
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117
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The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$
Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.
Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.
Let $G\triangleq\{\varphi\in L^2[...
- 139
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302
views
Banach space of discontinuous functions on a product space
Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-...
- 356
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179
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Non-reflexive linear subspace
We know that if X infinite dimensional normed space, then weak topology smaller than normed topology.
This is my problem(from russian textbook of Bogachev-Smolyanov, Functional Analysis) :
Let X be ...
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0
answers
151
views
Notion of solution of pde
Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in L^\infty(0,...
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145
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A question about the duality principle
Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by $$\Phi_Kf(y)=...
- 133
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35
views
Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T
Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...
- 345
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148
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existence of locally translation-invariant Borel measure on Frechet manifolds
It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
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137
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SVD of Frechet derivative
This is mainly a reference request. Is there a particular characterization of operators A from a Hilbert space H to itself such that the Frechet derivative A'(u) exists for each $u \in H$ and for any ...
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113
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Existence of a mapping in a nonseparable Banach space
Do there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that
$$
\forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad (...
- 1
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0
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182
views
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ \frac{j}{2^{...
- 1
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0
answers
77
views
How to generalize balanced and absorbing sets to R-modules?
I'm looking for generalizing the notions of balanced set and absorbing set. The goal is using them for analyzing topological R-modules with R being a unit ring.
It's easy to generalize balanced and ...
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153
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extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 1,835
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103
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The trivility of Besov space for large parameter
For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define
$$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$
and
$$W^{...
- 369
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0
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79
views
Stable analytic manifold under simple action
For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
- 1
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145
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Does there exist this special kind of homeomorphism?
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
- 183
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135
views
uniform continuity of a function in ultrametric spaces
Consider $[0,1]$ with the metric $d_1(x,y)=\left\{\begin{array}{cc}
0&x=y,\\
\max\{x,y\}&x\ne y.
\end{array}\right.$. Moreover let $(M,d_2)$ be an ultrametric space. Let
$f:(M,d_2)\to([0,1],...
- 1
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votes
0
answers
184
views
Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
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213
views
Equivalent Gaussian measures
Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, T is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. Let $...
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200
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Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$
We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$
I want to show for $r \geq 2$ that
$$\...
- 1
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votes
0
answers
606
views
$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in $C^0(...
- 266
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0
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88
views
References for LWP of a NLS Equation
I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...
- 516
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0
answers
206
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About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103