# Questions tagged [fa.functional-analysis]

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

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• 6,094
82 views

### Noether's theorem in the critical heat equation

I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me. I am watching a serie of lectures on "Blow up solution ...
• 318
1 vote
102 views

### Construction of the Lipschitz function with a given Lipschitz constant and given two values

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $g(b)=f(b)$ and Lipschitz ...
• 97
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### Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded

Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard ...
• 266
46 views

### The initial sigma-algebra on the dual of a Banach lattice

Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
• 131
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### General form of bounded linear functionals on Banach spaces

It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$. On p.188 of Introductory ...
• 619
1 vote
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### Find a finite partition of unity in the centralizer of a von Neumann algebra

Let $M$ be a von Neumann algebra and $\varphi$ is a faithful normal state on $M$. Suppose that $M_{\varphi}$ is a type II$_1$ factor. Suppose we have the following conclusion: if $p$, $q$ are any two ...
• 281
142 views

### The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
53 views

### Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?

Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
206 views

### A pexiderization of the sine addition law on semigroups

Can we solve the follwing functional equation $$f(xy)=g(x)h(y)+g(y)h(x)$$ on semigroups for unknown complex valued functions $f,g,h$ ?
1 vote
185 views

### Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
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112 views

### On a core for Neumann Laplacian on $C(\overline{D})$

Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
• 499
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### Deduce that a function is zero on interval $[0,M]$

I have been thinking about this for the last few days but I was not able to produce a definitive answer. Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
40 views

### What are general Horizon functions?

I was reading a paper from arxiv. There I found this definition of horizon functions Definition 2.2. Let $d \in \mathbb{N}_{\geq 2}$. Given any function $b:[0,1]^{d-1} \rightarrow[0,1]$, define the ...
• 79
1 vote
197 views

### On a property of complex exponentials

Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that $$\int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
• 3,301
153 views

### Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?

Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
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### Sufficient conditions on the density of a random vector $X$ to ensure that the density of $X^\top w$ is Holder-continuous for all $w \in \mathbb R^d$

Let $X$ be a random vector $\mathbb R^d$, with density $f$. For any any unit-vector $w \in \mathbb R^d$, let $\rho_w$ be the density of the random vector $X^\top w$. Question 1. Under what minimal ...
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• 3,421
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### Selecting some linearly independent columns of a particular matrix

Let us consider the matrix $C=A_1+A_2$ where : $A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$ $A_2$ is the the $n$ by $n$ block ...
• 3,421
284 views

### Do these properties characterize Hilbert spaces?

Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ isometric with $l_2^2$ such that $x\in E$. Does this property characterize a (...
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1 vote
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• 169
1 vote
196 views

### Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?

Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
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58 views

### A solution satisfying an integral inequality is bounded [closed]

Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality $$y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2}$$ ...
• 11
1 vote
168 views

### Finding $W^{1,\infty}$ solutions to an integral equation by fixed point

Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation \begin{align*} u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s. \end{...
• 75
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Let $\Lambda$ be a positive-definite matrix of size $n$ and let $R \ge 0$, which may depend on $n$. Consider the set $S := \{x \in \mathbb R^n \mid \|x\|_2 \le R,\,\|x\|_{\Lambda^{-1}} \le 1\}$ where $... • 6,094 1 vote 0 answers 51 views ### Properties of spectrally defined Sobolev spaces Let$\overline{M}$be a compact manifold with smooth boundary, and consider the Laplace operator$\Delta: H_0^1(M) \to H^{-1}(M)$and its inverse$\Delta^{-1}: H^{-1}(M) \to H_0^{1}(M)$. Then$\Delta^{...
Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$, \begin{align} \...