All Questions
12,935 questions
2
votes
0
answers
111
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 825
7
votes
1
answer
531
views
Conformal Killing fields satisfy a third order PDE
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
- 969
3
votes
1
answer
114
views
Dimension of spectral projection subspaces under local convergence
I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under ...
- 633
2
votes
1
answer
172
views
Does there exists an example of a Banach space that is compactly LUR; but not LUR
We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
- 85
0
votes
0
answers
126
views
Building representation of an arbitrary umbral calculus
Consider a set of integrable functions on the interval $(0,1)$.
Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).
In such system the ...
- 10.1k
1
vote
1
answer
115
views
Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator
I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time.
Consider the following ...
- 187
0
votes
0
answers
211
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
3
votes
1
answer
245
views
PDE: compactness vs blowup
There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following:
Solve (easier) approximate problems, show some form of compactness for the approximate ...
3
votes
2
answers
183
views
Dimension of spectral projection subspaces under strong convergence of operators
I have a possibly simple question regarding estimating bounds on spectral projection subspace.
Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...
- 633
5
votes
1
answer
267
views
Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions
Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
- 63
4
votes
0
answers
116
views
Weakly null sequences in projective tensor products II
The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1].
Below is a complimentary salad/side dish that accompanies the main course.
Let $B^2(X,Y)$ denote ...
- 2,605
1
vote
1
answer
120
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 141
0
votes
0
answers
112
views
Fourier integral operators and parametrix
Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary.
Question: Is there an expression for the ...
- 593
2
votes
1
answer
185
views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
3
votes
0
answers
204
views
Question about the formula of Green function of Laplacian on sphere
I'm reading a paper which said that
the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
- 809
2
votes
0
answers
114
views
Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 537
0
votes
1
answer
102
views
Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
- 505
1
vote
0
answers
108
views
Infinite tensor product of Hilbert spaces [duplicate]
Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
- 11
4
votes
1
answer
287
views
Reference request: Gaussian measures on duals of nuclear spaces
I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...
- 721
3
votes
1
answer
128
views
Fréchet-valued symbols
Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
- 395
1
vote
0
answers
89
views
Definition of second quantization
The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write:
Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
- 1,305
6
votes
1
answer
319
views
How are coordinate charts constructed in noncommutative geometry?
In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...
- 593
4
votes
1
answer
175
views
Large JN-sets in Banach spaces
For every infinite-dimensional Banach space $X$ there is a weak*-null sequence in the unit sphere of $X^\ast$. Does this extend under suitable circumstances to the non-separable setting?
Say that $X$ ...
- 41
0
votes
1
answer
192
views
A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 3,104
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
1
vote
1
answer
109
views
Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
1
vote
1
answer
162
views
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
7
votes
2
answers
627
views
Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
- 1,171
3
votes
0
answers
132
views
Takesaki's duality in representation theory of $C^*$-algebras
In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...
- 139
1
vote
2
answers
260
views
Interpretation of the singular integral for the definition of fractional Laplacian in classical case $s=1$
Fractional Laplacian is often defined via next principal value integral (assume dimension one for the simplicity)
$$(-\Delta)^su(x):=c_sP.V.\int_\mathbb{R} \frac{u(x)-u(y)}{|x-y|^{1+2s}}\, dy.$$
This ...
0
votes
0
answers
88
views
Intersection of Sobolev Spaces
Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\...
- 807
1
vote
1
answer
159
views
Existence of solution to nonlinear first order PDE with C^1 bounds
I'm looking for general existence of a PDE of the form
$$ f: U \times [0, \delta) \to \mathbb{R}$$
$$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$
where $f(p,0)$ is prescribed and $F$ is non-linear ...
- 337
0
votes
0
answers
63
views
Existence of a measurable maximizer
Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
- 1
0
votes
0
answers
72
views
Some questions about the concept of stable solution of elliptic PDE
For $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is stable if
$$
Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall ...
- 809
1
vote
1
answer
99
views
How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let
$$
\begin{matrix}
F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\
(x,z,p) \mapsto F(...
- 217
3
votes
2
answers
280
views
Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421
0
votes
1
answer
252
views
Reference request: log Sobolev inequality for uniform measure (uniform distribution over discrete set)
Suppose that $N \in \mathbb N_+$ is fixed and denote by $\mu = (\mu_0,\ldots,\mu_N)$ the uniform distribution on the set $\{0,1,\ldots,N\}$ (i.e., $\mu_n = \frac{1}{N+1}$ for each $0\leq n\leq N$). I ...
- 730
3
votes
0
answers
161
views
Lebesgue measure of the boundary of the positivity set of a function is zero?
Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $C=\{w>0\}$;
$w$ is subharmonic ...
0
votes
0
answers
126
views
A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
5
votes
0
answers
899
views
Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?
Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...
- 181
0
votes
0
answers
157
views
Dependence of functional integral on the function space
In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...
- 593
1
vote
0
answers
61
views
$L^p$ estimates for critical heat equation on $\mathbb{R}^n$
Background:
Let $u:\mathbb{R}^n\times [0,T_+)\to \mathbb{R}$ be a solution to
$$\partial_t u = \Delta u + |u|^{p-1}u$$
where $p=2^{*}-1=\frac{2n}{n-2}-1=\frac{n+2}{n-2}$, $n\geq 3$ and $T_+>0$ ...
- 537
0
votes
1
answer
415
views
Necessary conditions for convergence of convolution
In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
- 61
0
votes
0
answers
59
views
Series representation of functions
Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let
$$
V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\}
$$
...
- 93
0
votes
1
answer
131
views
Controlling convolutions with maximal functions
For $f\in L^1(\mathbb R^n),$ let $Mf$ be the (Edited: changed the type of maximal function) Stein spherical maximal function. Let $\varphi\in C_c^\infty.$ Then, can we have a pointwise estimate of the ...
- 1,755
1
vote
0
answers
43
views
Behaviour of higher order Laplacian in punctured domain
Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
- 11
0
votes
0
answers
55
views
Question on the modelling of (viscous) fluid in a bag with holes
Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)
What is the corresponding PDE to model the ...
- 1,334
0
votes
0
answers
46
views
Conservation law for generic linear hyperbolic PDEs?
Consider the wave equation:
$$
u_{tt} = \Delta u, \text{ on }U\times [0,T], u=0\text{ on }\partial U\times[0,T].
$$
To prove the only solution for the zero initial condition is zero, we only need to ...
- 1,755
2
votes
0
answers
78
views
Array-determined operator ideals
For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology).
An array in the Banach space $X$ is a sequence of sequences, $(...
- 121
2
votes
1
answer
197
views
Topology of ${\mathcal D}(\Omega)$ (space of test functions)
I have seen two approaches to the topology of ${\mathcal D}(\Omega)$:
(i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...
- 91