Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

learn more… | top users | synonyms (1)

0
votes
1answer
126 views

Countably generated $\sigma$-algebra

Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a standard probability space? I assume not, so here is a more ...
13
votes
2answers
449 views

A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that $\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
1
vote
1answer
102 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...
1
vote
0answers
61 views

inverse problem to resolution of the identity

Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define ...
0
votes
0answers
57 views

weakly precompact operators

A subset $S$ of $X$ is said to be weakly precompact provided that every sequence from $S$ has a weakly Cauchy subsequence. An operator $T:X\to Y$ is called weakly precompact (or almost weakly ...
7
votes
1answer
308 views

The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...
7
votes
1answer
241 views

What does the unique mean on weakly almost periodic functions look like?

There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
1
vote
0answers
58 views

Perturbation in Besov space

$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$. Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
6
votes
1answer
224 views

Completion of spaces of measures w.r.t. weak norms

For the sake of concreteness denote by $M_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x_0\in X$. On this space ...
0
votes
0answers
111 views

Extension of harmonic function with bounded $L^{2}$ norm

Let $h:D\setminus \{0\}\rightarrow \mathbb{R}$ be a harmonic function, where $D$ is the unit disc in $\mathbb{R}^{2}$, with bounded $L^{2}$ norm, i.e. $||h||_{L^{2}(D)}^{2}=\int_{D}|h|^{2}(x,y)dxdy ...
2
votes
0answers
32 views

Global Euclidean Carleman Estimate with a linear phase

I am interested in deriving the following global Carleman estimate which I think should hold : $ \| e^{\tau \phi} \triangle u \|_{L_{\delta}^2({\mathbb{R^3})}}> C \tau \| e^{\tau \phi} u ...
3
votes
0answers
59 views

Modify the jump set of $BV$ function

Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
4
votes
1answer
157 views

Linear extension operators for smooth functions: from compact sets to compact sets

I'm considering a situation where I have the linear restriction map of Fréchet spaces $$ C^\infty(C_1) \to C^\infty(C_2) $$ where $C_2 \hookrightarrow C_1$ are a pair of compact, connected subsets ...
0
votes
0answers
51 views

Lp space and zero order Besov space

I want to ask a basic question(may stupid), does the following relation holds: $$\|f\|_{B^0_{p,p}}\approx\|f\|_{L^p}$$ where, $\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the ...
23
votes
0answers
497 views

Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...
0
votes
0answers
96 views

Alternative representation of $C_c(X)$ as inductive limit

CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear. Under some additional constraints on the space (e.g. $X$ ...
6
votes
1answer
191 views

Simplicity of eigenvalues

Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
6
votes
3answers
219 views

On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator?

Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$. We say that $f\in C(K)$ is a generator of $C(K)$ when the set ...
4
votes
1answer
100 views

Every self-adjoint trace class operator on $L^2$ has integral kernel

I have asked this question on MSE but did not receive an answer. I thought I could try it here. Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be ...
2
votes
1answer
119 views

Visualizing ANOVA Decomposition [closed]

Let $f \in L^2[0,1]^d$ be a measurable function where $d \in \mathbb{N}$. For a given subset $u \subseteq D := \{1,2,\ldots,d\}$ consider the projections $P_u : L^2[0,1]^d \to L^2[0,1]^{|u|}$ given ...
1
vote
2answers
242 views

$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$

$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$. Here,Banach-space isomorphism means a bounded invertible operator ...
11
votes
2answers
234 views

Asymptotic behavior of Sturm-Liouville eigenvalues

I have two questions. Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$. Let $(\lambda_n)$ denote the sequence of eigenvalues of ...
3
votes
2answers
176 views

Bergman norm on a bigger domain

Let $D$ be a unit disc (I am actually interested in a much more general setting, but let's start with explicit examples). Let $E$ be an open subset of $D$. Consider the functional on the space of all ...
0
votes
2answers
70 views

Level sets and integral of functions of two variables

Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...
0
votes
1answer
75 views

$C^{\infty}_{loc}$-convergence - right definition

Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
2
votes
0answers
62 views

When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?

It would be soo awesome if you could help! For $ p \in \mathbb{N}$ consider the following $\mathcal{S(H)}^{p\times p}$-matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p}$ of nuclear operators ...
1
vote
0answers
78 views

About hyperplane separation theorem

I read in Lovasz's notes about semidefinite programs and combinatoric optimization. If $x_1A_1 + ... + x_nA_n\succ 0$ has no solution, then the linear subspace $L = x_1A_1 + ... + x_nA_n$ is ...
2
votes
1answer
179 views

Weak* extreme points

Suppose $x$ is an extreme point of the unit ball of a Banach space $E$. Embed $E$ in $E^{**}$ in the standard way. Is $x$ an extreme point of the unit ball of $E^{**}$?
1
vote
1answer
75 views

$L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X)$ for $X \subset Y$ dense?

is the Inclusion stated in the title true? In my case the spaces (essentially) are $X = H^1(\Omega)$ and $Y = L^2(\Omega)$ for $\Omega \subset \mathbb{R}$ bounded. My first try was to show $\lim_{t_1 ...
1
vote
0answers
113 views

Interchanging integrals and continuous linear forms in RKHS

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan. In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
0
votes
0answers
65 views

Fourier transform in Lorentz spaces

Denote by $E$ the class of nonnegative, even functions on $\mathbb{R}$, monotone decreasing to 0 on $\mathbb{R}_+$. In here, Y. Sagher stated the following proposition about Fourier transform of ...
6
votes
3answers
235 views

Non-self adjoint Sturm-Liouville problem

I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form: $(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} ...
1
vote
0answers
31 views

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too. Consider the following ...
8
votes
3answers
285 views

Cyclic vectors for the shift operator

Let $S:\ell^2\to\ell^2$ be the shift operator $$ S(x_1,x_2,\dots)=(0,x_1,x_2,\dots). $$ Let $x\in \ell^2$ with $x_1=1$. Is $x$ cyclic for $S$? In other words, is the span of the vectors ...
4
votes
1answer
99 views

For self-adjoint $T$ on $L^2(\mathbb{R}^n)$, when does $(1 + |x|)^{-1} (T - i \varepsilon)^{-1}(1 + |x|)^{-1}$ have a limit as $\varepsilon \to 0$?

Let $T : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be a possibly unbounded self adjoint operator. Let $R_\varepsilon$ denote the resolvent $(T - i \varepsilon)^{-1}$, $\varepsilon > 0$. Suppose ...
1
vote
1answer
116 views

trace of a commutator [duplicate]

Can one give an example of two bounded operators A and B in a Hilbert space such that both products AB and BA are of the trace class but their traces are different? If one of them is compact then the ...
5
votes
3answers
263 views

The mean of points on a unit n-sphere $S^n$

A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$ The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the ...
3
votes
1answer
76 views

Urysohn type cut off function

I am looking for a cutoff function. The Urysohn's Lemma says Let $X$ be a $T_{4}$ space and $A,B \subset X$ be two closed and disjoint subsets of $X$. Then there exists a continuous function $f:X ...
2
votes
0answers
74 views

Finite dimensional subrepresentations in a Fréchet space

The Lie group $G := SO(n, 1)$ acts on the hyperbolic space $\mathbb{H}^n$ by isometries. In particular, we have a representation of $G$ on $F := C^\infty(\mathbb{H}^n)$. I assume that $F$ is endowed ...
1
vote
3answers
143 views

Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]

Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in ...
5
votes
1answer
162 views

Eigenspace of a specific operator

Consider the operator $T:\ell^\infty({\mathbb N})\to\ell^\infty({\mathbb N})$ defined by $$ (Tx)_m=\sum_{k=m+1}^\infty p_{k,m} \ \ x_k, $$ where $$ p_{k,m}=\frac k{(k-1)(k-m)(k-m+1)}. $$ Then $T$ is a ...
3
votes
1answer
115 views

Stone-Weierstrass, uniform convergence, and sums

Assume $\{F_n\}_{n\in \mathbb{N}} \subset C(X)$ is some series of continuous complex functions on the compact Hausdorff space $X$. Assume also that the $F_n$ separate the points of $X$, and that the ...
4
votes
0answers
171 views

Compactly supported distributions as a projective G-module

For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
0
votes
0answers
65 views

Boundedness of a function that satisfies a PDE-type inequality

Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$. Suppose that $$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + ...
2
votes
1answer
137 views

Using $H^2$ to find a cyclic vector in $\ell^2$

Let us consider $\ell^p(\mathbb{Z})$. We know that the vector $e_1=(\dots,0,0,1,0,0,\dots)$ is a cyclic vector in sense that given the right shift operator $S:(\dots,x_0,x_1,x_2,\dots)\mapsto ...
3
votes
0answers
114 views

The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment. Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...
8
votes
2answers
305 views

Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad ...
2
votes
0answers
112 views

How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of $$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$ $$\partial_\nu u = \alpha \quad \text{on ...
11
votes
1answer
179 views

Weak*-closure of finite rank operators on dual space

Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is ...
2
votes
1answer
64 views

If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?

Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact. Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$. Is then $u \in L^\infty(0,T;X_1)$? To apply ...