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

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125 views

$Ax=b$ in a function space (again)

Let $X$ be compact Hausdorff topological space, $C(X)$ denote the algebra of complex-valued continuous functions on $X$, $b\in \mathbb{C}^m$, $\mathbf{A}\in C(X)^{m\times n}$, Let ${\mathbb{C}}^n$ ...
3
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1answer
118 views

$\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$. Does the following inequality (or something similar hold) for ...
1
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0answers
76 views

Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...
0
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0answers
68 views

Dipole Transition Integrals - Acceleration Form, What's Wrong?

I should have posted this question in a physics forum, but I think by posting in MathOverflow I may get more responses. The following question may sound stupid, since I'm sure I was wrong somewhere, ...
2
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1answer
268 views

Banach space of discontinuous functions(Killing continuous functions)

Edit: According to the comment of Prof. Majer, I revise the question: For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$ ...
2
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1answer
210 views

A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$. For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms ...
3
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1answer
268 views

When $C(X)$ is an injective $C(X)$-module?

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space. I would like to know if the weakened module version of this question is answered. More precisely: ...
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27 views

continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
3
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0answers
92 views

Continuously dependent on parameters [closed]

How do we check whether the solution is continuouly dependent on parameters? Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...
4
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2answers
169 views

Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?

I'm interested in a question regarding the identification of some duals of quasi-Banach spaces. However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the ...
0
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1answer
150 views

$Ax=b$ in a function space

Let $X$ be compact Hausdorff topological space, $C(X)$ denote the algebra of complex-valued continuous functions on $X$, $b\in \mathbb{C}^m$, $\mathbf{A}\in C(X)^{m\times n}$, for all $x\in X$, ...
2
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1answer
109 views

Question regarding to approximate continuity

Given $u\in BV(R^N)$, we say $u$ is approximate continuous at $x$ and the approximation limit is $l\in R$ if $$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$ for all ...
5
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2answers
154 views

Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
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0answers
70 views

Inverse problem to solve out current in the radiation problem

This is the radiation problem, but more like a analysis one. Suppose the current $f(x)\in L^2(\mathbb{R})$ has a compact support in $[-a,a]$. And the frequency $F(\theta)$ is given by ...
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0answers
157 views

Separability of the space $C(C[0, 1], \mathbb{R})$

Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$. I am wondering that ...
10
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2answers
480 views

Banach space modulo a one-dimensional subspace =?

My question is the following: Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed ...
0
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1answer
121 views

Equivalence of two definitions of Sobolev spaces

Good morning, I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space $$ D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) ...
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0answers
68 views

faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles. Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$. ...
3
votes
2answers
128 views

Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...
7
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1answer
246 views

Bounded operator on a normed space with empty spectrum

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space. Is there any example of bounded operator acting on a ...
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0answers
82 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...
2
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0answers
98 views

Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point. I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...
5
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1answer
141 views

Reflexive subspaces of non-separable abstract $L_1$ spaces

An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...
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0answers
117 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 ...
17
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0answers
252 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
0
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0answers
34 views

proving that a complicated function is concave or strongly uni-modal

I am trying to prove the concave property for a complicated function during my research project (imperfect maintenance modelling for starter) which has the following form: $\eta(t)= \alpha \beta ...
3
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1answer
99 views

Example of joint cyclic and separating vector

Let $\mathcal{H}$ be a separate Hilbert space and $\mathcal{B}(\mathcal{H}) \subset \mathcal{B}(\mathcal{H}) \otimes M_2(C)$ be a W$^*$-inclusion pairs. It is known that this pair share a joint cyclic ...
3
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0answers
43 views

Behaviour of Markov type under uniform homeomorphism of spheres

A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if, for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has $$ ...
0
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1answer
207 views

Continuity of a Functional

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is ...
3
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0answers
94 views

Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in ...
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0answers
44 views

decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$ Let $j\in \mathbb N$ ...
0
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0answers
38 views

Bound on change of function given bound on Hessian

Suppose I have very some smooth function $F(x)$, and let $x_0 = \text{argmin}_x F(x)$. I would like to bound $F(x) - F(x_0)$ from above, in terms of the gradient $\nabla f(x)$ and the Hessian matrix ...
9
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3answers
303 views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
3
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4answers
310 views

Continuity in Banach space for non-linear maps

I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...
3
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0answers
132 views

Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback. Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...
7
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0answers
272 views

Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} ...
0
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0answers
104 views

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 ...
2
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0answers
97 views

What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it. Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi ...
2
votes
1answer
156 views

Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...
4
votes
1answer
207 views

C*-Algebras: Dynamics vs. Derivations

Problem Given a C*-algebra $\mathcal{A}$. Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$. (More precisely, strongly continuous ...
1
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1answer
161 views

On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
0
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1answer
254 views

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...
4
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1answer
217 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
2
votes
1answer
247 views

Sobolev Space, “characteristic function” for the weak derivative

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product $$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$ for $u\in H_0^1$ and ...
4
votes
2answers
620 views

What does analyticity imply in complex analysis? [closed]

In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not ...
9
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1answer
404 views

Between compact and locally uniform: What is the name of this convergence?

Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f_n)$ a sequence in $Y^X$ with the following property: For every $x_0\in X$ and every $\varepsilon>0$, there exist a ...
1
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2answers
289 views

How to prove that a kernel is positive definite?

For example, how to prove $\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$ where $\sigma > 0$ is a parameter, is positive definite? I have tried to ...
2
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1answer
226 views

Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
4
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1answer
301 views

When is a `1-form' with continuous coefficients exact?

Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in ...
0
votes
1answer
157 views

Orthogonal projection

Let $G$ be an operator with compact resolvent on a Hilbert space $H$ such that $\ker G \neq \{0\}$. Further let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} := G+P$. My question is: ...