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6 votes
1 answer
510 views

Path integral as quantum mechanics on the tangent bundle

Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a ...
Dmitry Vaintrob's user avatar
6 votes
1 answer
118 views

Infra-Pták space that is not Pták

From reading the literature of the 1970s heyday of locally convex spaces, it seems that it was an important open question whether there is an infra-Pták (i.e. $B_r$-complete) space that is not Pták (i....
Cameron Zwarich's user avatar
6 votes
1 answer
320 views

Stability of fractional Sobolev spaces under diffeomorphisms

Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
Vincent's user avatar
  • 83
6 votes
1 answer
339 views

Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$

Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to ...
MCS's user avatar
  • 1,284
6 votes
1 answer
119 views

Tensoring adjointable maps on Hilbert modules

Given a right Hilbert $A$-module $E$, and a right Hilbert $B$-module $F$, together with non-degenerate $*$-homomorphism $\phi:A \to \mathcal{L}_B(F)$, we can form the tensor product $$ E \otimes_{\phi}...
Dave Shulman's user avatar
6 votes
2 answers
251 views

uniform approximation by a particular set of functions

Consider the interval $[0,1]$ and let $\mu_k(t)$ with $k=1,\ldots,n$ be continuous functions such that they are all strictly increasing on the interval $[0,1]$ and such that $\mu_1(t)<\mu_2(t)<\...
Ali's user avatar
  • 4,115
6 votes
1 answer
173 views

local property of nonlocal differential operators

Motivation: In general, a nonlocal operator acting on the product of two functions doesn't have the product rule as the local operator does. However, the Hilbert transform on the real line has a very ...
Ivy Hsu's user avatar
  • 78
6 votes
1 answer
579 views

The elliptic regularity theorem for differential operators with variable coefficients

I'm following the book "Introduction to the theory of distributions" by Friedlander and Joshi. There is the following result p. 109 Theorem (8.6.1). Let $X \subset \mathbb{R}^n$ be an open set, and ...
Andrew's user avatar
  • 589
6 votes
1 answer
271 views

Is the Mackey topology $\tau(l^{\infty},l^{1})$ strongly Lindelöf?

Let $l^{\infty}$ (respectively, $l^{1}$) be the space of bounded (respectively, absolutely summable) real sequences. I need to find out if $l^{\infty}$ equipped with the Mackey topology $\tau(l^{\...
OzE's user avatar
  • 63
6 votes
1 answer
478 views

Decay of positive definite function in $L^p$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means $$ \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i-x_j)y_i y_j \geq 0 $$ for all $...
Linden's user avatar
  • 217
6 votes
1 answer
1k views

Tensor product of measure spaces

For a compact topological space $X$, denote by $\mathcal{M}(X)$ the Banach space of finite signed Borel (Radon) measures on $X$ with the total variation norm. This is canonically isometric to the dual ...
Matthias Ludewig's user avatar
6 votes
2 answers
457 views

$L^{p}$ isoperimetric inequalities on the Hamming cube

Let $A \subset \{-1,1\}^{n}$ be a subset of the Hamming cube with cardinality $|A|=2^{n-1}$. Define $w_{A} : \{-1,1\}^{n} \to \mathbb{N}\cup \{0\}$ so that $w_{A}(x)$ to be number of boundary edges ...
Paata Ivanishvili's user avatar
6 votes
1 answer
321 views

Derivatives of norm of vector-valued holomorphic functions

Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
erz's user avatar
  • 5,529
6 votes
1 answer
570 views

Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...
Kate Juschenko's user avatar
6 votes
1 answer
378 views

Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
Hamed's user avatar
  • 105
6 votes
1 answer
2k views

Comparing norms on tensor products of matrices

Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$ where $(s_1,...
R.N's user avatar
  • 209
6 votes
2 answers
371 views

Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side: $$ \Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial t}+\beta^2w\right)+\...
Mechanical engineer's user avatar
6 votes
1 answer
215 views

Boundary values of boundary value problems

Let $M$ be a manifold with smooth boundary. We can consider the Dirichlet or the Neumann problem on $M$. Let $(\phi_k)$ be an orthonormal basis of eigenfunctions to the Dirichlet problem and let $(\...
Matthias Ludewig's user avatar
6 votes
1 answer
737 views

Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in sup-norm)....
AlexE's user avatar
  • 2,998
6 votes
3 answers
450 views

Positivity of the top Lyapunov exponent

I have a general question about the Oseledets Multiplicative Ergodic Theorem. In the context of the MET I'd like to know if there is some reasonably general sufficient condition which implies that the ...
Ilya Kapovich's user avatar
6 votes
1 answer
697 views

Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to $$ -\Delta u=f\hspace{3cm}(1)? $$ I'm of ...
leo monsaingeon's user avatar
6 votes
3 answers
758 views

Decomposition of $L^2(\Gamma \backslash G)$

Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ (...
Pig's user avatar
  • 809
6 votes
2 answers
799 views

A question on density of Lipschitz functions in weighted Sobolev spaces

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$. Let now $...
Changyu Guo's user avatar
  • 1,881
6 votes
1 answer
680 views

Is there an operator algebraic reformulation of the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
Sebastien Palcoux's user avatar
6 votes
2 answers
794 views

Plancherel formula for non-second-countable (non-unimodular) groups

The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstract Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a ...
The User's user avatar
  • 2,442
6 votes
1 answer
423 views

Absolutely 2-summable operator on a Hilbert space

An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that $$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \right)^{1/2} ...
Matthias Ludewig's user avatar
6 votes
1 answer
249 views

What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the $L^...
sd234's user avatar
  • 173
6 votes
1 answer
498 views

A finely open set, not open up to polar set?

I already asked this on M.SE, but get no answers. Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
gerw's user avatar
  • 1,724
6 votes
3 answers
1k views

functional subrings

I should recall the notion of maximal subring of a commutative unitary ring $R$. Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
Ali Reza's user avatar
  • 1,788
6 votes
2 answers
460 views

Terminology: Banach spaces equipped with continuous associative product?

This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach algebra/...
6 votes
2 answers
487 views

Induction theorems for finite-dimensional complex representations of infinite groups

Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
Fabian Lenhardt's user avatar
6 votes
2 answers
293 views

quasinilpotence and finite spectrum

Let A be a quasinilpotent operator on a Hilbert space and let $A^{*}A$ have finite spectrum. Does then follow, that A is nilpotent ?
jjcale's user avatar
  • 2,753
6 votes
1 answer
476 views

Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant?

Context: I am currently reading through the freely available lecture notes from Tristan Riviere (here) on the applicability of integration by compensation in the analysis of various geometrically ...
Glen Wheeler's user avatar
6 votes
1 answer
453 views

The typical size of a random element in a Banach space

Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
Tom LaGatta's user avatar
  • 8,512
6 votes
1 answer
427 views

Subspaces of $L^{2}$

[In what follows $0^{0}$= 1 by convention.] Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$ such that $\left\lvert f\right\rvert^{\left\lvert f\right\rvert}$ belongs to $...
Ady's user avatar
  • 4,060
6 votes
1 answer
170 views

Do projections in an $AW^\ast$-algebra form an orthomodular lattice?

I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
David Gao's user avatar
  • 2,830
6 votes
1 answer
292 views

Analytic maps on Banach spaces: analyticity upgrade

Consider the following problem. Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and $$ f:U\to G $$ an analytic map, such ...
Lorenzo Pompili's user avatar
6 votes
1 answer
251 views

Finite representability of $\ell_p$ in subspaces of $L_p(0,1)$

Let $M$ be a closed subspace of $L_p(0,1)$, $1<p<\infty$, $p\neq 2$. Suppose that M contains copies of $\ell_p^n$ uniformly. Does $M$ contain a copy of $\ell_p$? The result is true for $p=1$, ...
M.González's user avatar
  • 4,461
6 votes
1 answer
177 views

Properties of non-integer powers of the Hodge Laplacian

Consider a complete smooth Riemannian manifold $(M,g)$. I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...
V. Moretti's user avatar
6 votes
1 answer
174 views

Sobolev space is spanned by distributions supported on half-lines?

I asked this question on Mathematics Stack Exchange previously. This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it. For any $s \leq 1/2$, $$H^s(\mathbb{...
Tian Xia's user avatar
6 votes
1 answer
193 views

Oscillatory integrals with a decaying factor in the integrand

Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased): Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
Evandra's user avatar
  • 63
6 votes
1 answer
827 views

Best way to introduce B-splines?

I have the option of mentoring pure math undergrads in a topic lying within Approximation Theory and I really want to do $B$-splines. Mostly because I have recently found applications of them in my ...
Dan1618's user avatar
  • 197
6 votes
2 answers
499 views

When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?

If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
apanpapan3's user avatar
6 votes
1 answer
256 views

Perron-Frobenius and Markov chains on countable state space

The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer: Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
Landauer's user avatar
  • 173
6 votes
1 answer
575 views

English translation of Schwartz's papers on vector-valued distributions

I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
genfuntranslate's user avatar
6 votes
2 answers
392 views

Another reference for higher order Fourier analysis

I am trying to read Tao's Higher order Fourier analysis but I would be very happy to find another book on the subject. I would like to learn something about the Gowers norm and about Roth's theorem (...
user avatar
6 votes
1 answer
360 views

Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that: If $\lambda\gg 1$...
Patch's user avatar
  • 377
6 votes
1 answer
204 views

Smoothness of family of distributions

Let $X$ be a compact manifold. Denote by $\mathscr{D}^\prime(X \times X)$ the space of tempered distributions on the cartesian product $X \times X$. Given two test functions $\varphi, \psi \in \...
Matthias Ludewig's user avatar
6 votes
1 answer
196 views

Circular sequences continuous?

I noticed something interesting when playing around with Mathematica. Consider the sum $$x(N)= \frac{1}{N^2} \sum_{i=1}^{N-1} \frac{1}{1-\cos(2\pi i/N)}$$ this sequence will converge to $1/6$ as $N$...
user avatar
6 votes
1 answer
549 views

Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
sharpe's user avatar
  • 721

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