**-2**

votes

**0**answers

70 views

### Normal structure

Let $X$ be a Banach space and $C$ a closed, convex, bounded and non-trivial set of $X$. It is said that $C$ has normal structure if for each closed, convex, bounded and non-trivial set $B\subset C$ ...

**4**

votes

**0**answers

102 views

### $L_\infty(\mu)$ spaces non-isomorphic to a dual space

Given a measurable space $(\Omega,\mu)$ such that $L_\infty(\mu)$ is isomorphic to a dual space, $L_\infty(\mu)$ is an injective Banach space. Indeed, given a subspace $Y$ of $X$ and a norm-one ...

**2**

votes

**2**answers

134 views

### Composition operators on fractional-order (periodic) Sobolev spaces

(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...

**0**

votes

**1**answer

57 views

### Sobolev chain rule on non-compact manifolds

Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions ...

**1**

vote

**1**answer

115 views

### Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable?

I have the following Question:
1) Is it true that
if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ implies $\int\limits_{\Omega}{e^vdx}=+\infty$ ...

**2**

votes

**1**answer

110 views

### Bounding $\int_{\mathbb{R}_{+}}W(x,\lambda)W(y,\lambda)d\lambda$

Let $$W:\mathbb{R}_{+}^{2}\rightarrow(0,1]$$ be a symmetric, integrable function. Let $$f(x)=\int_{\mathbb{R}_{+}}W(x,\lambda)d\lambda$$ and assume this function is monotonically non-increasing. Is ...

**5**

votes

**0**answers

137 views

### Non-linear positive map

In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...

**1**

vote

**0**answers

31 views

### Degenerate Carleman Estimate for Laplace Beltrami Operator

Suppose $(M,g)$ is a compact Riemannian manifold with boundary and that I have been able to prove a Carleman type estimate of the following form for an explicit phase function $\phi$ :
$ \| e^ {\tau ...

**0**

votes

**0**answers

99 views

### Is there a rather natural space an automorphism of which is the Mellin transform?

Disclaimer: this question might be a little too vague and thus not suitable for this site despite the soft-question tag. If so, feel free to migrate it to MSE.
I just read this and, trying to find ...

**0**

votes

**0**answers

38 views

### Mapping sphere surface to a vector space such that distances are preserved? [migrated]

I have a unit radius sphere (say in 3D) centered in origin. Thus the shortest distance between two points on the sphere is the geo-desic. Is there a transformation (linear or non-linear) on the points ...

**1**

vote

**1**answer

197 views

### The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator

Consider a general bilinear multiplier operator:
$$
T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta,
$$
where $\Pi$ is the torus, $n\in\mathbb{Z}$, ...

**4**

votes

**2**answers

153 views

### Is taking the product of signed measures weakly continuous?

For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...

**0**

votes

**0**answers

40 views

### Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$

Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...

**4**

votes

**1**answer

112 views

### Injectivity of the Funk transform for nonsmooth functions

Let $S^{n-1}$ be the unit sphere in $\mathbb R^n$ and $\Gamma_n$ the collection of great circles on it.
Assume $n\geq3$.
The Funk transform of a function $f:S^{n-1}\to\mathbb R$ is a map ...

**1**

vote

**1**answer

113 views

### Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is ...

**0**

votes

**0**answers

98 views

### Unitarizability of group representations

Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...

**0**

votes

**1**answer

93 views

### Square Integrable Harmonic Functions in an Infinite Strip

Suppose $S= \left\{x \in \mathbb{R}^3 : a <x_1< b \right\} $ is an infinite strip the three dimensional Euclidean Space.
Is it true that the only $L^2$ harmonic function in this strip is the ...

**0**

votes

**0**answers

60 views

### What does the Plancherel theorem say about positive-definite distributions?

I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.
The ...

**2**

votes

**0**answers

32 views

### Nemytskii/superposition operator without separability of Banach space?

Let $T:[0,1] \times X \to \mathbb{R}$ be a nonlinear map where $X$ is a Banach space. Suppose that $T$ is a Caratheodory map, so that $t \mapsto F(t,x)$ is measurable and $x \mapsto F(t,x)$ is ...

**0**

votes

**1**answer

180 views

### On Cantor sets every map is $C^{\infty}$ [closed]

For a fixed Cantor set $K\subset [0,1]$ and a continuous function $g:[0,1]\to \mathbb R.$ Is it always possible to find a $C^{\infty}$ map $f:[0,1]\to \mathbb R$ such that $g$ and $f$ coincide in $K?$
...

**6**

votes

**0**answers

177 views

### Does hypoellipticity imply the existence of a parametrix?

Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...

**4**

votes

**2**answers

312 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**4**

votes

**1**answer

103 views

### Differential Operators On A Curve And On Osculating Circle

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...

**0**

votes

**1**answer

130 views

### About weak derivatives [closed]

I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...

**4**

votes

**1**answer

242 views

### A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $

Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...

**2**

votes

**1**answer

256 views

### The Tensor product of algebra group

Let G is a locally compact group. Is the following true?
The tensor product of $L^1(G)$ with $L^1(G)$ is $L^1(G \times G)$.

**-1**

votes

**2**answers

320 views

### What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...

**0**

votes

**1**answer

76 views

### Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...

**3**

votes

**0**answers

66 views

### Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...

**1**

vote

**0**answers

91 views

### Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?

Let $G$ be a Lipshitz domain and $u_n \to u$ in $W_p^1(G)$. Is it correct, that $\frac{\partial |u_n|}{\partial x_m} \to \frac{\partial |u|}{\partial x_m}$ in $L_p(G)$?
I know, that $\frac{\partial ...

**0**

votes

**0**answers

59 views

### solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...

**4**

votes

**2**answers

166 views

### Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ...

**6**

votes

**2**answers

345 views

### Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity?
$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$
Here, $|x|$ denotes the pointwise absolute ...

**0**

votes

**0**answers

25 views

### Writing eigen functions of one Stochastic Process in terms of the eigen functions of another

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...

**0**

votes

**0**answers

34 views

### Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...

**0**

votes

**0**answers

81 views

### Differentiating and integrating an infinite series arising from a PDE

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $(\varphi_k, \lambda_k)$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Given $u \in H^{\frac ...

**3**

votes

**2**answers

170 views

### Discretizing probability measures

Consider a probability distribution on $\mathbb{R}^k$, say $\mu$. Then there is a sequence of probability measures $\mu_n$ that converge weakly to $\mu$ so that each of them is discrete (takes ...

**2**

votes

**2**answers

151 views

### Extremal functions for Gagliardo-Nirenberg inequality

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$(∫|u|^{r}dx)^{\frac{1}{r}} \leq ...

**2**

votes

**1**answer

72 views

### Is this series involving hyperbolic functions uniformly convergent?

Suppose that
$\mu_k$ is an increasing sequence of numbers such that $0 < \mu_1 \leq \mu_2 \leq ..$ with $\mu_k \to \infty$ as $k \to \infty$
$\sum_{k=1}^\infty |u_k|^2 < \infty$ and ...

**2**

votes

**1**answer

173 views

### An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold
$\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...

**2**

votes

**1**answer

105 views

### Conformally covariant distributions

In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form
$$
\langle O(x)O(y)\rangle,
$$
where $O$ is a scalar primary field. Scale covariance ...

**1**

vote

**1**answer

89 views

### 'Test Functions' to Lower Bound the Norm of Elements of Dual Quantum Group

There may well be an answer to this question in a simpler category than that of finite dimensional quantum groups and in that case this question is more suitable to math.stack and I apologise in ...

**0**

votes

**1**answer

64 views

### Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...

**0**

votes

**1**answer

97 views

### How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
...

**3**

votes

**0**answers

58 views

### An estimate for the maximal C* norm in the group algebra of a free group

Let F--->G be an epimorphism of groups, F being finitely generated and free. Let H be its kernel. Consider a lifting i: G--->F of the epimorphism. Every element of C[G] is of the form
a=sum a(g) i(g) ...

**1**

vote

**0**answers

63 views

### When does analytic in the operator norm imply analytic in the trace class norm?

This is a crosspost from MSE. It's been up there for a few weeks now. A 200 rep bounty yielded no results (or even comments). I'm hoping someone here has some helpful ideas. See this post for the ...

**2**

votes

**1**answer

74 views

### Domain of fractional powers of operators

Let $A$ and $B$ be non-negative ($(A x, x) \geq 0$ for all $x \in \mathcal{D}(A)$, similarly for $B$) densely defined self-adjoint operators on a Hilbert space $H$. Then the spectral theorem defines ...

**0**

votes

**1**answer

102 views

### Uniform convergence of Fourier (orthonormal) expansion of series

Let $u \in L^2(M)$ on some closed Riemannian manifold. We can write
$$u = \sum_{k \geq 0}(u,\varphi_k)\varphi_k$$
if $\varphi_k$ is some o.n basis of $L^2$ with is orthogonal in $H^1$ (eg. ...

**1**

vote

**0**answers

26 views

### Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...

**1**

vote

**1**answer

160 views

### Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...