All Questions
1,154 questions
3
votes
1
answer
476
views
Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
- 175
3
votes
2
answers
470
views
If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer ...
- 1,396
3
votes
1
answer
383
views
"Nice" functions on infinite-dimensional space of germs of continuous functions at a point
Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
- 24.9k
3
votes
3
answers
1k
views
Classification of a certain System of Linear First Order PDEs whose characteristic polynomial has one real and two complex conjugate zeros
In my research work, I recently have come across the following system of three linear first order pde's whose characteristic polynomial has one real and two complex conjugate zeros.
$3u_{1,1}+u_{3,1}...
- 573
3
votes
1
answer
769
views
Strichartz estimates for the heat equation
Consider the heat flow $e^{t\Delta}$, $t>0$, on an Euclidean domain (say $\mathbb{R}^3$). I expect, in analogy with the Strichartz estimates for the Schrodinger equation, that the following ...
- 943
3
votes
1
answer
368
views
Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?
Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...
3
votes
2
answers
869
views
How do functions operate in a Sobolev space $H^{s}$?
Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ ...
- 1,051
3
votes
1
answer
225
views
Quotient of weak amenable Banach algebras
Let $A$ be a weak amenable Banach algebra and $I$ be a closed (two-sided) ideal of $A$. In general $\frac{A}{I}$ is not weakly amenable. Is there an example of this type of weak amenable Banach ...
- 565
3
votes
1
answer
518
views
Connection between the Fourier transform of f and |f|
If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
\|\widehat{f}\|_{L^{p'}}\...
- 425
3
votes
1
answer
845
views
Moser estimates?
Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
- 31
3
votes
0
answers
233
views
Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
3
votes
1
answer
274
views
Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements
Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...
- 366
3
votes
2
answers
948
views
Can one hear the shape of a drum for operators?
M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...
- 1,583
3
votes
1
answer
329
views
Survey on functional equations and inequalities
Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
user60665
3
votes
0
answers
144
views
Transitive closure of balanced bounded mass transport
Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
- 19.7k
3
votes
2
answers
1k
views
Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
- 3,738
3
votes
1
answer
476
views
Kernel of the Radon transform
Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density (...
- 21.8k
3
votes
0
answers
252
views
Existence of solutions to a reaction-diffusion problem.
Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\...
- 193
3
votes
0
answers
443
views
Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?
My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call $...
- 1,199
3
votes
1
answer
775
views
Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?
Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.
But do we have any quantitative ...
- 1,893
3
votes
1
answer
428
views
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...
- 3,477
3
votes
0
answers
338
views
Method of characteristic for a system of first order PDEs
I am working with this system of first order PDEs:
\begin{equation}
\left\{
\begin{aligned}
%Suscettibili
&\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
- 31
3
votes
0
answers
376
views
is this a known method for solving PDEs
I recently posed a system of PDEs to solve on MSE at https://math.stackexchange.com/q/514147/36530. It was quickly solved by a nice pair of subsitutions.
However, in this post, I'd like to show here ...
- 453
3
votes
0
answers
306
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
3
votes
0
answers
154
views
Elliptic estimates in unweighted Sobolev spaces
In several sources (Choquet-Bruhat & Christodoulou 1981, Nirenberg-Walker 1973) estimates for elliptic partial differential equations on a noncompact manifold are derived in weighted Sobolev ...
- 419
3
votes
0
answers
208
views
Analytic solution to two component, first order, linear PDE system
I would like to obtain analytic solutions to the following PDE system:
\begin{equation}
\rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1}
\end{equation}
with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
- 268
3
votes
1
answer
949
views
ODE continuous dependence on parameters to PDE
I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
- 55
3
votes
1
answer
402
views
Concavity near the boundary of the distance function
I was reading the paper
Quelques remarques sur les problemes elliptiques quasilineaires du
second ordre, P. L. Lions, Journal d’Analyse Mathématique volume 45,
pages 234–254(1985)
and on page 251 he ...
- 375
3
votes
1
answer
311
views
The distribution of roots of elliptic polynomial
If $p(x)$ is an $n$ variables polynomial of even degree with complex coefficients which satisfies the strong elliptic condition, that is, Re$p(x) \ge C|x|^{2m}$ for any $x \in \mathbb R^n$ where $2m$ ...
- 1,441
3
votes
1
answer
2k
views
leray schauder fixed point and schauder fixed point
I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...
- 54
3
votes
2
answers
804
views
Non symmetric coefficient matrix for elliptic PDE
Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form
$$ D_i(a_{i,j}D_ju)=0 \tag{1}$$
where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
- 265
3
votes
1
answer
251
views
Null sets in PDE
Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$,
$$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = \...
- 193
3
votes
1
answer
1k
views
Duality map in strictly convex Banach spaces
Say $(B,\|\cdot\|)$ is a finite dimensional, strictly convex Banach space. Is it true that the map $\phi:B^*\rightarrow B$ which takes a linear functional $f$ with $\|f\|=1$ into the unique unit norm ...
- 39
3
votes
0
answers
411
views
Bounded functions dense in Sobolev Spaces
Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
- 9,853
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
3
votes
1
answer
1k
views
A calculus question related to the nonnegative definite functions
I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
- 1,649
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
3
votes
1
answer
463
views
"Strongly mutually singular" families of measures, and the set of ergodic measures
Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume ...
- 708
3
votes
0
answers
160
views
Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras
I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
3
votes
0
answers
300
views
Distance to the level sets of an almost linear function
While reading about the almost-splitting theorem of Cheeger and Colding, I thought about the following question, which asks about wether the level sets of a function which is close to being a linear ...
- 4,123
3
votes
1
answer
740
views
Centralizers in C*-algebra
Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$?
For ...
- 179
3
votes
1
answer
365
views
Final topology of surjective linear map on Banach space
Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...
- 315
3
votes
2
answers
265
views
Can one realize this as an ergodic process?
Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.
We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$
In other words: For ...
user135480
3
votes
1
answer
788
views
Higher order functional derivatives
Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ ...
- 3,535
3
votes
1
answer
233
views
Extending linear isometries from subspaces of $\ell_p^n$
Take $p\in (1,\infty)\setminus \{2\}$. Let $X$ be a subspace of $\ell_p^n$ and let $U\colon X\to \ell_p^m$ ($m\geqslant n$) be a linear isometry. Is it possible to extend $U$ to a (non-surjective) ...
- 331
3
votes
1
answer
479
views
Extension of positive functionals II
This is a follow-up to
Extension of positive functionals.
Assume that $X=R^n$ with the canonical order (I indicate with $K$ the positive cone, $x \in K$ iff $x_i \ge 0$ for all $i$) and let $L:M \to R$...
- 6,035
3
votes
2
answers
5k
views
Finite speed of propagation of wave equation
Let us consider the wave equation $$(\partial_t^2 - \Delta)u = 0$$ on a domain $U$ with coercive homogeneous boundary conditions $$Bu|_{\partial U} = 0$$ that make $-\Delta$ self-adjoint. My question ...
- 33
3
votes
1
answer
695
views
Continuous embedding between parabolic Sobolev spaces
I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?
Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
- 503
3
votes
1
answer
975
views
Distributional derivative is locally integrable - then the distribution as well?
Given a distribution $T \in D'(\mathbb{R})$ such that the distributional derivative $\partial T \in L^1_{loc}(\mathbb{R})$. Can one deduce that $T \in L^1_{loc}(\mathbb{R})$ as well? Or can anyone ...
- 403
3
votes
1
answer
142
views
Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
- 839