All Questions
10,240 questions
6
votes
1
answer
191
views
Fractional derivative notation in wave turbulence
This is my first question in MathOverflow and I will do my best to format it correctly and make it clear.
I am reading a paper on dispersive wave turbulence which introduces the following family of ...
- 163
6
votes
1
answer
182
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,115
6
votes
1
answer
645
views
Bochner-Minlos for moment-generating functions?
It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have ...
- 661
6
votes
1
answer
378
views
Optimal constant in Sobolev embedding
It is well-known that the Sobolev space $H^1(0,s)$ embeds continuously in the space of continuous functions $C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is $\sqrt{\...
- 3,388
6
votes
1
answer
509
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 ...
- 7,962
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....
- 3,816
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 ...
- 83
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}...
- 969
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)<\...
- 4,115
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 ...
- 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^{\...
- 63
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 ...
- 9,853
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 ...
- 3,946
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 ...
- 5,529
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 ...
- 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| $$
where $(s_1,...
- 209
6
votes
2
answers
371
views
Weak solutions for a PDE of fourth order
I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side:
$$
\Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial t}+\beta^2w\right)+\...
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 $(\...
- 9,853
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)....
- 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 ...
- 421
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 ...
- 5,380
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 $...
- 1,881
6
votes
1
answer
233
views
Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$
Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:
$$
\| T_t : L_p(\mu) \to L_q(\mu)\| \...
- 1,090
6
votes
1
answer
680
views
Is there an operator algebraic reformulation of the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
6
votes
1
answer
422
views
Absolutely 2-summable operator on a Hilbert space
An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that
$$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \right)^{1/2} ...
- 9,853
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^...
- 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 ...
- 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 ...
- 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 ...
- 976
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 ?
- 2,753
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 ...
- 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 $...
- 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 $H$,...
- 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 ...
- 1,075
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 $p=1$, ...
- 4,461
6
votes
1
answer
342
views
The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds
(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
- 221
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, ...
- 205
6
votes
1
answer
173
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 $s \leq 1/2$,
$$H^s(\mathbb{...
- 86
6
votes
1
answer
827
views
Best way to introduce B-splines?
I have the option of mentoring pure math undergrads in a topic lying within Approximation Theory and I really want to do $B$-splines. Mostly because I have recently found applications of them in my ...
- 197
6
votes
2
answers
499
views
When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?
If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
- 199
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....
- 173
6
votes
1
answer
574
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 & ...
- 183
6
votes
2
answers
391
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 (...
user147650
6
votes
1
answer
403
views
Interpolation for Sobolev spaces
How one can identify the following (complex) interpolation space
$$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$
where $\Omega$ is a regular domaine. After research, it seems that ...
- 395
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 \...
- 9,853
6
votes
1
answer
182
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence of ...
- 6,741
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 ...
- 721
6
votes
1
answer
580
views
Why is it difficult to solve the Monge problem directly?
I'm trying to understand something about the Monge problem. The Monge problem is:
Let $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
- 427
6
votes
1
answer
212
views
Nice S¹-action implies existence of unconditional basis?
Let $V$ be a Banach space equipped with a continuous linear action of $S^1$ (meaning, the map $S^1\times V\to V$ is continuous). Assume that all the eigenspaces of the $S^1$-action are finite ...
- 43.2k