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Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
MyShepherd's user avatar
7 votes
0 answers
306 views

Gottfried Helms' tetra-eta series

Here Gottfried Helms introduces the following fascinating divergent series $$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$ The terms don't go to zero, so technically the series does not converge ...
Caleb Briggs's user avatar
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198 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 ...
Aareyan Manzoor's user avatar
7 votes
0 answers
204 views

Global generation of $S^n \Omega_X$ for a fake projective plane

Let $X$ be a fake projective plane, namely, a compact complex surface with $$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample. Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
Francesco Polizzi's user avatar
7 votes
0 answers
120 views

What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?

Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
alesia's user avatar
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193 views

Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements

Consider the free product of $\mathbb{Z}/2$ with itself with generators $$ \mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle $$ and regard its group $C^*$-algebra $$ C^*(\mathbb{Z}/2*\mathbb{...
C-star-W-star's user avatar
7 votes
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2k views

Algebraizing topology and analysis via condensed mathematics

I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow. I've just come across a Twitter thread by Laurent Fargues explaining a work ...
Ythyb's user avatar
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0 answers
168 views

The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space

Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
 V. Rogov's user avatar
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317 views

Multiple Fourier series

In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem: Theorem 1: The trigonometric series $$\...
Elmustapha NADIR's user avatar
7 votes
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129 views

holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)

Let $M$ be a Stein manifold with smooth, strictly pseudoconvex boundary, and $x$ a point on its boundary. Is there a holomorphic function $f$ on $M$, smooth on the boundary, with strict maximum of $|f|...
Misha Verbitsky's user avatar
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207 views

Is the derivative the unique operation on points in the plane that preserves convexity?

Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$ Such that the ...
Chris H's user avatar
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203 views

Polynomials having all zeros in the closed left half plane

Let $$P(z) =\sum_{k=0}^n(\alpha_k+e^{i\gamma}\beta_k)z^k=P_1(z)+e^{i\gamma}P_2(z)$$ be a polynomial of degree $n$ with $\alpha_k, \beta_k\geq 0$ for $0\leq k\leq n, $ where $$P_1(z) =\sum_{k=0}^n\...
user159888's user avatar
7 votes
0 answers
348 views

Zero derivative on a connected set

I apologize in advance for this (not really research level) question whose answer should be well known. Complex differentiability of a function $f:A\to\mathbb C$ for $A\subseteq \mathbb C$ without ...
Jochen Wengenroth's user avatar
7 votes
0 answers
169 views

Limiting behavior of a sequence of polynomials

Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line $\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary fact (equivalent to Lemma 9.13 here) that if $u\in\mathbb{C}$ and $|u|=1$, ...
Richard Stanley's user avatar
7 votes
0 answers
242 views

Has this Banach algebra been studied?

Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm ...
Alan's user avatar
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769 views

How much differs the category of real-analytic manifolds from $C^\infty$ ones?

I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
John117's user avatar
  • 395
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0 answers
132 views

Smoothing property of a certain singular integral operator of non-convolution type

For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
Matt Rosenzweig's user avatar
7 votes
0 answers
351 views

Fractional Laplacian and chain rule

For the classical Laplacian, we have $$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$ for smooth functions $h$ and $u$. Does a similar chain rule hold (up to a reminder term) also for the ...
Zac's user avatar
  • 161
7 votes
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294 views

Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$ For general locally ...
Jannik Pitt's user avatar
  • 1,474
7 votes
0 answers
158 views

$C^*$ algebras whose nontrivial projections form a non empty compact connected set

Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set? Is there an example of this situation such that ...
Ali Taghavi's user avatar
7 votes
0 answers
353 views

If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?

I hope this question isn't too basic or ambiguous for this site. The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$: $$\left[\begin{matrix}p & ...
wlad's user avatar
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7 votes
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3k views

Definition of homogeneous Sobolev spaces

As we know the inhomogeneous Sobolev space (we only consider $s>0$) $${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
Slm2004's user avatar
  • 633
7 votes
0 answers
107 views

Potential p-norm on tuples of positive operators

This is a follow-up to this question on p-norms of tuples of operators. Consider $\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$, meaning $A,B\geq 0$, and define $$ \left\|\left[\begin{...
Chris Ramsey's user avatar
  • 3,984
7 votes
0 answers
124 views

The bidual of the space of divergence-free vector fields

Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
Tomasz Kania's user avatar
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7 votes
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420 views

What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?

Here is the story as I see it. Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
Tim Phalange's user avatar
7 votes
0 answers
248 views

Isometries on the unit sphere

Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
Markus's user avatar
  • 1,361
7 votes
0 answers
237 views

Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
  • 1,903
7 votes
0 answers
373 views

What is known about "almost orthogonal vectors"?

Motivation: Suppose we have a kernel $k(a,b)$ defined over the natural numbers. Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
user avatar
7 votes
2 answers
824 views

Fourier series of smooth functions in infinitely many variables

Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
Boris Tsygan's user avatar
7 votes
0 answers
85 views

Analogue of Friedrichs extension for Hilbert $C^*$-modules

Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
geometricK's user avatar
  • 1,903
7 votes
0 answers
461 views

On a paper of Alain Connes entitled 'Around Wilson's Theorem '

A relatively recent paper Alain Connes - Around Wilson's theorem introduced the function $$ S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr). $$ In the same paper, he proved that the ...
user avatar
7 votes
0 answers
177 views

Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?

I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
Yemon Choi's user avatar
  • 25.8k
7 votes
1 answer
394 views

Inverse limit in the category of $C^{\ast}$-algebras or operator spaces

Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces? I tried to search but could not find a proper reference. Any reference or comments about ...
Math Lover's user avatar
  • 1,115
7 votes
0 answers
432 views

(geodesic) smoothness of f-divergence with respect to the Wasserstein metric

We consider the f-divergence, which takes the form $$ D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ. $$ For example, when $f(t) = t \log t$, we obtain the KL-divergence. My question is ...
Minkov's user avatar
  • 1,127
7 votes
0 answers
181 views

Compact Kaehler submanifolds of projectivized Hilbert space

If we take a separable complex Hilbert space $H$, its projective space $PH$ is an infinite-dimensional Kähler manifold in a fairly obvious sense (see below). Suppose $M \subset PH$ is a finite-...
John Baez's user avatar
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7 votes
0 answers
327 views

Status of two Banach space theory open problems posted by Pełczyński

In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems. Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is ...
Idonknow's user avatar
  • 623
7 votes
0 answers
200 views

Equivalent strictly convex norms in spaces of small density

Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm? Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a ...
Tomasz Kania's user avatar
  • 11.3k
7 votes
0 answers
243 views

Loomis-Whitney versus Gagliardo inequalities

When searching for a reference, I discovered a curious fact about the Wikipedia page concerning the Loomis-Whitney Inequality (LWI).This page, which exists only in an English version, states that the ...
Denis Serre's user avatar
  • 52.3k
7 votes
0 answers
619 views

Lavrentiev Phenomenon

Does there exist a (onedimensional) integral functional of calculus of variations $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that not only $$ \inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
Carlo Mantegazza's user avatar
7 votes
0 answers
222 views

Can C*/W*-algebras be realized as (involutive?) monoid/co-monoid objects?

I would like to know how close one can get to realizing the category of C*-algebras as a category of monoid objects. Related (almost, but not quite, duplicate) questions are: "Recovering a monoidal ...
Tom Mainiero's user avatar
7 votes
0 answers
549 views

Counter-example to the completeness of the Wasserstein metric

$\newcommand{\P}{\mathcal{P}}$ Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
Oleg's user avatar
  • 931
7 votes
0 answers
478 views

Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"

For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization $$ L^p (\mu) = \left\{f : X \to \Bbb{...
PhoemueX's user avatar
  • 734
7 votes
0 answers
1k views

Books on von Neumann algebras

I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
Mathbuff's user avatar
  • 455
7 votes
0 answers
132 views

Different definitions of fractional sobolev spaces

Let $\Omega$ be a bounded and smooth domain in $\mathbb R^d$. For any $s\in (0,1)$ we can define $H_s(\Omega)$ to be the space of functions $u\in L^2(\Omega)$ such that $$(x,y)\mapsto \frac{|u(x)-u(y)|...
Thomas's user avatar
  • 630
7 votes
0 answers
198 views

Does this bound on an average over character sums have a more direct proof?

A special case of a well known result of Ingham is that $$\sum_{n\leq x} d(n)d(n+1)=\frac{6}{\pi^2}x(\log x)^2+O(x\log x)$$ where $d(n)$ is the number of divisors of $n$. Ingham's results, which ...
Kevin Smith's user avatar
  • 2,480
7 votes
0 answers
3k views

What is vague convergence and what does it accomplish?

For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
Greg Zitelli's user avatar
  • 1,104
7 votes
0 answers
187 views

distance distributions on a hypersphere?

Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define $$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$ where ...
T. Amdeberhan's user avatar
7 votes
0 answers
452 views

Sufficient condition on coefficients for a complex power series to be bounded

Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form $$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$ where $c_{2n}\geq 0$ for all $n$. Is there a ...
Deepti's user avatar
  • 783
7 votes
0 answers
304 views

Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let $d\in\left\{2,3\right\}$ $\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
0xbadf00d's user avatar
  • 167
7 votes
0 answers
501 views

intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
R Mary's user avatar
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