Questions tagged [fa.functional-analysis]

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

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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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1 answer
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 \begin{equation} y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2} \end{equation} ...
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1 vote
1 answer
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{...
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1 vote
1 answer
75 views

Tight upper-bounds for the Gaussian width of intersection of intersection of hyper-ellipsoid and unit-ball

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 $...
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1 vote
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52 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^{...
1 vote
0 answers
83 views

How to prove a concentration isoperimetric inequality for a non-Lipschitz function

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} \...
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3 votes
2 answers
428 views

Do two ways to differentiate Lipschitz functions coincide?

Let $f\colon \Omega \to \mathbb{R}$ be a Lipschitz function on an open subset $\Omega\subset \mathbb{R}^n$. By the Rademacher theorem $f$ has first derivative almost everywhere. We denote it $\nabla f$...
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1 vote
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56 views

Reference request: rates of weak convergence of Polish space-valued random variables

Let $(E,\mathscr{E})$ be a Polish space $E$ together with its Borel $\sigma$-algebra $\mathscr{E}$. Let $(\Omega, \mathscr{F},P)$ be a probability space and let $(X^{(n)})_{n \in \mathbb{N}}$ be a ...
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4 votes
1 answer
179 views

Are nonatomic probability measures on a Banach space nicely shrinking a.e?

Let $\mu$ be a nonatomic probability measure on a Banach space $X$. Is it true that for $\mu$ a.e. $x \in X$, the function $g_x: (0, \infty) \to \mathbb R$ given by $$g_x (r) := \mu(B_r (x))$$ is ...
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2 votes
1 answer
153 views

On the infimal convolution of two norms on $\mathbb R^n$

$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let \begin{equation*} K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1), \end{equation*} \begin{equation*} M:=M_{n,...
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1 answer
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Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?

Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f_a(x) := \|x-a\|_2 + \|x\|$ is strictly convex ...
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3 votes
0 answers
117 views

Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?

I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
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0 answers
47 views

Concentration inequality for inner product of two Lipschitz functions

I was reading chapter 5 of the book HDP(Roman Vershynin). There I find theorem 5.1.4 extremely fascinating. I am curious to discover does this theorem hold to the inner product of two Lipschitz ...
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2 votes
1 answer
113 views

Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?

For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by $$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
3 votes
0 answers
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Conditions ensuring that the paraproduct remainder is well-defined

In short, my question is: are there conditions that one can impose on two tempered distributions $u$ and $v$ that will guarantee that the paraproduct remainder $R(u,v)$ is well-defined and is "...
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3 votes
0 answers
129 views

How to prove the polar decomposition of unbounded operators?

Let $ T $ be a closed, densely defined operator on a Hilbert space $ H $. Then there exists a positive self-adjoint operator $ A $, $ D(A)=D(T) $ and a isometric operator $ V:R(A)\to \overline{R(T)} $ ...
2 votes
1 answer
80 views

Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$

For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define $$ \begin{split} K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\ M_n(a,t) &:= ...
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2 votes
0 answers
45 views

Translation request: Boundedness of Cauchy integral on Lipschitz boundary

The reference: "L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes" (https://annals.math.princeton.edu/1982/116-2/p04) is written in French. Can we ...
0 votes
0 answers
68 views

Is there an eigenspace decomposition of the space of matrices that is induced from the underlying vector space?

Given a positive-definite hermitian metric $g^{i\bar{\jmath}}$ on the vector space $C^{n}$, there is an induced metric on the space $\mathcal S_n$ of symmetric hermitian matrices that is given by $$(A,...
5 votes
0 answers
175 views

If $\lim_{t\to +\infty} \int_{0}^{\pi} f(x)\exp(e^{xt}) \, dx=0$ then $f=0$ a.e?

Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and $$ F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt $$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is ...
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1 vote
0 answers
47 views

Maximum principle for fractional Laplacian

The classical maximum principle for harmonic functions goes as follows: Let $\Omega \subseteq \mathbb{R}^n$ be an open, bounded, connected set and let $u$ be a harmonic function on $\Omega$. Then $$\...
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1 vote
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58 views

Integration with respect to Haar measures normalised over a subspace

Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work $\int_{\mathcal{U}(d)} \frac{\...
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42 views

A certain property of positive-semidefinite infinite matrices

In this answer I concluded with this: For which arrays $\big(\sigma_{ij}\big)_{(i,j)\in\mathbb N^2}$ [of real numbers] whose every upper-left corner is positive-semidefinite does line $(1)$ above ...
1 vote
0 answers
58 views

"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$

Let us consider the Burgers equation $$u_t + (u^2)_x = 0$$ In Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
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2 votes
1 answer
185 views

Strategy of the proof of the "minimal entropy condition" for scalar conservation laws

Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law $$u_t + f(u)_x = 0,$$ satisfying the entropy condition $$\eta(u)_t + q(u)_x \le 0$$ in the ...
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1 vote
1 answer
119 views

Spectral perturbation theory of discrete spectra in presence of continuous spectrum

This is a 2 part question: 1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...
10 votes
1 answer
283 views

Maximal ideals of the ring $\mathbb C \{T\}$

Consider the Banach $\mathbb C$-algebra $$ \mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace $$ With the norm given by $\| \sum a_i T^i\| = \sum |...
5 votes
1 answer
174 views

Is the Borel lemma projection a smooth principal bundle?

Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map $$ J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty $$ returning the ...
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5 votes
0 answers
50 views

Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
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4 votes
2 answers
216 views

Is this an $L^p-L^{\infty}$ operator?

Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions: $$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t} \int_{|x-...
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5 votes
0 answers
215 views

Equality from the Grothendieck inequality

I asked the following question on math.stackexchange.com but have not received any response. So I would like to try my luck here. This question is related to the Grothendieck inequality. Let field $\...
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2 votes
0 answers
55 views

Under what conditions does a continuous linear map map a closed subspace to a closed subspace?

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$? It is obviously satisfied if $W$ is ...
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4 votes
0 answers
56 views

The embedding of a Banach lattice in an ultrapower

Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...
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2 votes
0 answers
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Existence of analytic function in disk algebra [closed]

Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
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3 votes
1 answer
240 views

Extremely disconnected or extremally disconnected?

In the context of Banach space theory, what is the correct terminology: extremally disconnected or extremely disconnected. Looking through the internet I have met using both extremely and extremally ...
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1 vote
0 answers
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When can we characterize Sobolev space $W^{2k,p}(\Omega)$ only via the Laplacian-like terms

One of the characterizations of the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$ uses the Fourier transform $\mathcal{F}$: $f \in W^{s,p}(\mathbb{R}^n)$ iff $f$ is a tempered distribution such ...
1 vote
1 answer
54 views

Is $\sup\{\|A\widehat{k}_{\lambda}\|: \lambda\in\Omega\}=\sup\{\|A^*\widehat{k}_{\lambda}\|: \lambda\in\Omega\}$? where $A$ is an operator on RKHS

A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....
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2 votes
0 answers
36 views

Green function of a 2D exterior domain

Consider solutions of the laplace equation \begin{equation} \begin{split} -\Delta u=f, \ \ u|_{\partial D}=0, \end{split} \end{equation} where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
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1 vote
0 answers
116 views

An estimate for the first eigenfunction of the fractional Laplacian and a kind of "maximum principle"

Let $\Omega$ be a bounded Lipschitz domain. Let $u$ be the first eigenfunction of the fractional Laplacian $$ (-\Delta)^s u = \lambda u \ \text{ in } \Omega, \quad u = 0 \ \text{ in } \mathbb R^n \...
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3 votes
0 answers
139 views

Rate of uniform approximation by piecewise constant functions

Definitions and Notation: Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$. For every positive integer $N$, define the ...
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0 votes
0 answers
60 views

Regularity of map from $L^p$ to $\mathcal{W}_p$

Let $(X,d)$ be a polish metric space, fix a probability measure $\mathbb{P}$ on $(X,d)$ belonging to the Wasserstein $\mathcal{W}_p(X,d)$ for some fixed $p\in [1,\infty)$. Denote the Borel $\sigma$-...
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0 votes
0 answers
114 views

Type III von Neumann algebra

Let $\mathcal M$ be a type $\mathrm{III}$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of these lecture ...
0 votes
0 answers
69 views

Matrix algebra as sub algebra of von Neumann algebra

Let $\mathcal M$ be a infinite dimensional von Neumann algebra which is not abelian. Suppose it is known that $\mathcal M$ does not contain type $I_n$ factors von Neumann subalgefbras for all $n\geq N....
1 vote
0 answers
153 views

Proving a sign rule for $f_{2n}$

If $t_{1},...,t_{n}$ are real numbers, consider the set of indexed linear operators $T(t_{1}),...,T(t_{n})$ on a Hilbert space $\mathcal{H}$ and define its ordering by: $$\pi[T(t_{1})\cdots T(t_{n})] :...
1 vote
0 answers
39 views

Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
0 votes
0 answers
241 views

How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...
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0 answers
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Is $\phi(t)=\|P(w+td)-w\|_X/t$ nonincreasing if $X$ is "only" a uniformly smooth and uniformly convex reflexive Banach space?

For a Hilbert space $X$ it is known that the function $\phi(t)=\frac{1}{t}\|P(w+td)-w\|_X$ with $t>0$ is nonincreasing. Here, $P:X\to C$ denotes the projection operator and $w \in C, d \in X$ are ...
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2 votes
0 answers
46 views

Explicit estimates on summability kernels

A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that $$ \int_0^1 k_n(t) \mathrm d t =1,$$ $$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant ...
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7 votes
1 answer
200 views

Square-root lattices: where do they appear?

As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
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1 vote
0 answers
53 views

A statement on completeness of complex exponentials

I'm currently reading a paper by Olevskii on almost integer translates: https://www.sciencedirect.com/science/article/pii/S0764444297878731 In this paper the author considers for a given sequence $\{ \...
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