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Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that \begin{equation} \sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
Paz's user avatar
  • 61
6 votes
0 answers
213 views

Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
Tutukeainie's user avatar
6 votes
0 answers
529 views

Infinite-dimensional "algebraic varieties"

This question was also formerly posted on MSE but has not received any answer or comment. Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
Zerox's user avatar
  • 1,543
6 votes
0 answers
208 views

Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain

I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...
Hannes's user avatar
  • 2,670
6 votes
0 answers
318 views

Is there any connection between deformation theory in algebraic geometry and perturbation theory in functional analysis/PDEs?

Particularly, is there any connection between formal/first-order/infinitesimal deformation theory and perturbation theory? Both subjects involve "perturbing" some structure at a point, so ...
xuq01's user avatar
  • 1,094
6 votes
0 answers
182 views

Factorization of metric space-valued maps through vector-valued Sobolev spaces

Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that $$ \int_{x\in X}\,d(y_0,f(x)...
ABIM's user avatar
  • 5,405
6 votes
0 answers
211 views

Regularity of $|u|^{\alpha}$ when $u$ is Schwartz

Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
Benjamin Pineau's user avatar
6 votes
0 answers
120 views

Condition on the support of $f$ which ensure that $\widehat{f}$ has a zero-measure nodal region

Suppose that $f\in L^2(\mathbb{R})$ is non-zero and compactly supported. Then its Fourier transform $\widehat{f}\neq 0$ is analytic, and in particular the nodal set $\{\xi\in\mathbb{R}\,s.t.\,\widehat{...
RaffaeleScandone's user avatar
6 votes
0 answers
1k views

Condensed/liquid vector spaces and path integrals

[Edited to take into account comments.] Background One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
curioser's user avatar
6 votes
0 answers
208 views

Can every weakly converging sequence be made to converge strongly after taking a subsequence and rearranging?

Let $f_i: [0, 1] \to \mathbb R$ be functions in $L^1 \cap L^\infty$ with $\sup_i \|f_i\|_{L^\infty} < M$ for some $M > 0$. Suppose $f_i$ converge weakly in $L^1$ to some $L^1$ function $f$ - ...
Nate River's user avatar
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6 votes
0 answers
290 views

Two questions about Fock spaces

Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
JustWannaKnow's user avatar
6 votes
0 answers
277 views

Homogenized Mehta integral

For $a=(a_1,a_2)$ and $b=(b_1,b_2)$ in $\mathbb{C}^2$ let me use the notation from classical invariant theory $$ (ab)=a_1b_2-a_2b_1\ . $$ If I run out of letters $a,b,c,\ldots$, then I will switch to $...
Abdelmalek Abdesselam's user avatar
6 votes
0 answers
124 views

Meagre sets of bounded operators

Let $H$ be a separable, infinite-dimensional Hilbert space and let $\mathbb{B}(H)$ be the algebra of bounded operators on $H$. The norm topolology on $\mathbb{B}(H)$ is stricly finer, hence the ...
Matthias Ludewig's user avatar
6 votes
0 answers
113 views

A continuity argument for a dispersive $gKdV$ estimate

I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at $$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$ where $F(u) = u^5$ (for example). The ...
David Bowman's user avatar
6 votes
0 answers
129 views

Weak-type inequality for the partial Fourier sum operator

I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark: For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
pureorapplied's user avatar
6 votes
0 answers
132 views

Mazur-Ulam bases in finite-dimensional Banach spaces

Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
Lviv Scottish Book's user avatar
6 votes
0 answers
107 views

Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE $$u'(t) = (A+B) u(t), \quad u(0)=u_0$$ which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
Sascha's user avatar
  • 536
6 votes
0 answers
92 views

Does every compact abelian group contain a Kronecker set generating a dense subgroup?

Let $G$ be a compact metrizable abelian group with infinite exponent. Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
Ethan Ackelsberg's user avatar
6 votes
0 answers
533 views

Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
TheWildCat's user avatar
6 votes
0 answers
153 views

How absolute is NIP in a model?

The following question is motivated by a model theoretic question but doesn't really involve any model theory per se. That said, I don't know the appropriate keywords for the relevant functional ...
James E Hanson's user avatar
6 votes
0 answers
241 views

Extension of positive functionals

Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...
Giorgio Metafune's user avatar
6 votes
0 answers
190 views

Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$

Lie algebraically, the eigenvalue of the spherical function \begin{align*} \phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*) \end{align*} ...
user48713's user avatar
6 votes
0 answers
445 views

Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. $\textbf{Theorem}$ Let $1\...
Tony419's user avatar
  • 421
6 votes
0 answers
107 views

Real-world example of a Banach *-algebra with a nonzero *-radical

Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
Cameron Zwarich's user avatar
6 votes
0 answers
158 views

Quotients of subspaces of $C(\alpha)$

A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
S Argyros's user avatar
  • 986
6 votes
0 answers
348 views

Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett

( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .) Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
user avatar
6 votes
0 answers
144 views

Does every locally convex space with a Schauder basis have the approximation property?

For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property. Since both the notion of Schauder bases and of the approximation property are well ...
Christian's user avatar
  • 799
6 votes
0 answers
99 views

Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?

The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature. Can every ...
Tomasz Kania's user avatar
  • 11.3k
6 votes
0 answers
315 views

Question on operator topologies convergence

Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of bounded operators on $H$. It is known that the strong operator topology and the norm topology on $\mathcal{B}(H)$ ...
javi1996's user avatar
  • 355
6 votes
0 answers
321 views

Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
Chilote's user avatar
  • 596
6 votes
0 answers
181 views

Blocksum induces a unital H-space structure on the space of Fredholm operators

Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
Eric Schlarmann's user avatar
6 votes
0 answers
239 views

Sheaves on Rectifiable Sets

Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory? Background: The jumping off point here is a simple analogy - geometric measure ...
Juan Sebastian Lozano's user avatar
6 votes
0 answers
113 views

Interpolation of some Sobolev spaces

Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$ I am wondering what is $$(X_0,X_2)_{1/2,2}=?$$ Would it be $H^2_0(0,...
Saj_Eda's user avatar
  • 395
6 votes
0 answers
117 views

Homomorphisms from BV

Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\...
Maciej Ciechowski's user avatar
6 votes
0 answers
237 views

A characterisation of certain $C^*$-algebras

I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
Mark Roelands's user avatar
6 votes
0 answers
921 views

Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle

I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature. I'm hoping there's a nice ...
MichaelGaudreau's user avatar
6 votes
0 answers
219 views

Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
Vesselin Dimitrov's user avatar
6 votes
0 answers
83 views

Are invertible measures strictly dense?

Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
Jan_Ch.'s user avatar
  • 113
6 votes
0 answers
88 views

Density of squares of radial eigenfunctions

The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
Joonas Ilmavirta's user avatar
6 votes
0 answers
282 views

Spectral properties of Non-local Differential operators on real line

I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs. Definition: A ...
mystupid_acct's user avatar
6 votes
0 answers
4k views

Interchange of supremum and integral

Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure. Are there some results under which the following interchange ...
yon's user avatar
  • 303
6 votes
0 answers
240 views

Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces

I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
Javier's user avatar
  • 493
6 votes
0 answers
203 views

Uniform estimates of Fourier transform of tempered functions with parameters

Consider the following function in $\mathbb{R}^3$: $$ f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2}, $$ where $\frac{1}{2}<\alpha<1$, and ...
Tomas's user avatar
  • 879
6 votes
0 answers
388 views

Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
Steve's user avatar
  • 1,095
6 votes
0 answers
184 views

Reference request: fusion rules for unitary dual of SL(3,R)?

By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
Yemon Choi's user avatar
  • 25.8k
6 votes
0 answers
137 views

Spectrum of perturbed differential operators

I am looking for a reference that could help me with the following two questions: Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential ...
Dominick's user avatar
6 votes
0 answers
486 views

On Fourier coefficients of nonnegative function

Let $N$ be nonnegative integer, $F(x)$ be a nonnegative real Lebesgue integrable function defined on $[0,1]$. Suppose that all Fourier coefficients $c(\lambda)=\int_0^1F(x)e^{-2\pi i \lambda ...
Alexey Ustinov's user avatar
6 votes
0 answers
281 views

Covariance operator analogue for manifolds and respective measure manifolds

Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
Nik Bren's user avatar
  • 519
6 votes
0 answers
798 views

What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?

Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
user avatar
6 votes
0 answers
210 views

Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$. The $L^p$ norm on $M$ is given by \begin{...
Rauan Akylzhanov's user avatar

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